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Preface to the First Edition. 

The new volume which I herewith offer to the mathematical public, 
and especially to the teachers of mathematics in our secondary schools, 
is to be looked upon as a first continuation of the lectures Uber den 
mathematischen Unterricht an den hoheren Schulen*, in particular, of those 
on Die Organisation des mathematischen Unterrichts** ]by Schimmack and 
me, which were published last year by Teubner. At that time our concern 
was with the different ways in which the problem of instruction can be 
presented to the mathematician. At present my concern is with deve- 
lopments in the subject matter of instruction. I shall endeavor to put 
before the teacher, as well as the maturing student, from the view-point 
of modern science, but in a manner as simple, stimulating, and con- 
vincing as possible, both the content and the foundations of the topics 
of instruction, with due regard for the current methods of teaching. 
I shall not follow a systematically ordered presentation, as do, for 
example, Weber and Wellstein, but I shall allow myself free excursions 
as the changing stimulus of surroundings may lead me to do in the 
course of the actual lectures. 

The program thus indicated, which for the present is to be carried 
out only for the fields of Arithmetic, Algebra, and Analysis, was indicated 
in the preface to Klein-Schimmack (April 1907). I had hoped then that 
Mr.. Schimmack, in spite of many obstacles, would still find the time to 
put my lectures into form suitable for printing. But I myself, in a way, 
prevented his doing this by continuously claiming his time for work in 
another direction upon pedagogical questions that interested us both. 
It soon became clear that the original plan could not be carried out, 
particularly if the work was to be finished in a short time, which seemed 
desirable if it was to have any real influence upon those problems of 
instruction which are just now in the foreground, As in previous years, 
then, I had recourse to the more convenient method of lithographing 
my lectures, especially since my present assistant, Dr. Ernst Hellinger, 
showed himself especially well qualified for this work. One should not 
underestimate the service which Dr. Hellinger rendered. For it is a 
far cry from the spoken word of the teacher, influenced as it is by 
accidental conditions, to the subsequently polished and readable record. 

* On the teaching of mathematics in the secondary schools. 
** The organization of mathematical instruction. 


In precision of statement and in uniformity of explanations, the lecturer 
stops short of what we are accustomed to consider necessary for a printed 

I hesitate to commit myself to still further publications on the 
teaching of mathematics, at least for the field of geometry. I prefer to 
close with the wish that the present lithographed volume may prove 
useful by inducing many of the teachers of our higher schools to renewed 
use of independent thought in determining the best way of presenting 
the material of instruction. This book is designed solely as such a mental 
spur, not as a detailed handbook. The preparation of the latter I leave 
to those actively engaged in the schools. It is an error to assume, as 
some appear to have done, that my activity has ever had any other 
purpose. In particular, the Lehrplan der Unterrichtskommission der Ge- 
sellschaft Deutscher Naturforscher und Arzte* (the so-called "Meraner" 
Lehrplan} is not mine, but was prepared, merely with my cooperation, 
by distinguished representatives of school mathematics. 

Finally, with regard to the method of presentation in what follows, 
it will suffice if I say that I have endeavored here, as always, to combine 
geometric intuition with the precision of arithmetic formulas, and that 
it has given me especial pleasure to follow the historical development 
of the various theories in order to understand the striking differences 
in methods of presentation which parallel each other in the instruction 
of today. 

Gottingen, June, 1908 


Preface to the Third Edition. 

After the firm of Julius Springer had completed so creditably the 
publication of my collected scientific works, it off erred, at the suggestion 
of Professor Courant, to bring out in book form those of my lecture 
courses which, from 1890 on, had appeared in lithographed form and 
which were out of print except for a small reserve stock. 

These volumes, whose distribution had been taken over by Teubner, 
during the last decades were, in the main, the manuscript notes of my 
various assistants. It was clear to me, at the outset, that I could not 
undertake a new revision of them without again seeking the help of 
younger men. In fact I long ago expressed the belief that, beyond a 
certain age, one ought not to publish independently. One is still 
qualified, perhaps, to direct in general the preparation of an edition, but 
is not able to put the details into the proper order and to take into proper 
account recent advances in the literature. Consequently I accepted the 

* Curriculum prepared by the commission on instruction of the Society of 
German Natural Scientists and Physicians. 

offer of Springer only after I was assured that liberal help in this respect 
would be provided. 

These lithographed volumes of lectures fall into two series. The 
older ones are of special lectures which I gave from time to time, and 
were prepared solely in order that the students of the following semester 
might have at hand the material which I had already treated and .upon 
which I proposed to base further work. These are the volumes on Non- 
Euclidean Geometry, Higher Geometry, Hyper geometric Functions, Linear 
Differential Equations, Riemann Surfaces, and Number Theory. In con- 
trast to these, I have published several lithographed volumes of lectures 
which were intended, from the first, for a larger circle of readers. These 

a) The volume on Applications of Differential and Integral Calculus 
to Geometry, which was worked up from his manuscript notes by 
C. H. Miiller. This was designed to bridge the gap between the needs 
of applied mathematics and the more recent investigations of pure 

b) and c) Two volumes on Elementary Mathematics from an Advanced 
Standpoint, prepared from his manuscript notes by E. Hellinger. These 
two were to bring to the attention of secondary school teachers of mathe- 
matics and science the significance for their professional work of their 
academic studies, especially their studios in pure mathematics. 

A thoroughgoing revision of the volumes of the second series seemed 
unnecessary. A smoothing out, in places, together with the addition of 
supplementary notes, was thought sufficient. With their publication 
therefore, the initial step is taken. Volumes b), c), a) (in this order) will 
appear as Parts I, II, III of a single publication bearing the title Ele- 
mentary Mathematics from an Advanced Standpoint. The combining, in 
this way, of volume a) with volumes b) and c) will meet the approval 
of all who appreciate the growing significances of applied mathematics 
for modern school instruction. 

Meantime the revision of the volumes of the first series has begun, 
starting with the volume on Non-Euclidean Geometry. But a more 
drastic recasting of the material will be necessary here if the book is 
to be a well-rounded presentation, and is to take account of the recent 
advances of science. So much as to the general plan. Now a few words 
as to the first part of the Elementary Mathematics. 

I have reprinted the preface to the 1908 edition of b) because it 
shows most clearly how the volume came into existence 1 . The second 
edition (1911), also lithographed, contained no essential changes, and 
the minor notes which were appended to it are now incorporated into 

1 My co-worker, R. Schimmack, who is mentioned there, died in 1912 at the 
age of thirty-one years, from a heart attack with which he was seized suddenly, 
as he sat at his desk. 


the text without special mention. The present edition retains 1 , in the 
main, the text of the first edition, including such peculiarities as were 
incident to the time of its origin. Otherwise it would have been necessary 
to change the entire articulation, with a loss of homogeneity. But during 
the sixteen years which have elapsed since the first publication, science 
has advanced, and great changes have taken place in our school system, 
changes which are still in progress. This fact is provided for in the 
appendices which have been prepared, in collaboration with me, by 
Dr. Seyfarth (Studienrat at the local Oberrealschule). Dr. Seyfarth also 
made the necessary stylistic changes in the text, and has looked after 
the printing, including the illustrations, so that I feel sincerely grateful 
to him. My former co-workers, Messrs. Hellinger and Vermeil, as well 
as Mr. A. Walther of Gottingen, have made many useful suggestions 
during the proof reading. In particular, I am indebted to Messrs. Vermeil 
and Billig for preparing the list of names and the index. The publisher, 
Julius Springer has again given notable evidence of his readiness to 
print mathematical works in the face of great difficulties. 

Gottingen, Easter, 1Q24 


Preface to the English Edition. 

Professor Felix Klein was a distinguished investigator. But he was 
also an inspiring teacher. With the rareness of genius, he combined 
familiarity with all the fields of mathematics and the ability to perceive 
the mutual relations of these fields; and he made it his notable function, 
as a teacher, to acquaint his students with mathematics, not as isolated 
disciplines, but as an integrated living organism. He was profoundly 
interested in the teaching of mathematics in the secondary schools, both 
as to the material which should be taught, and as to the most fruitful 
way in which it should be presented. It was his custom, during many 
years, at the University of Gottingen, to give courses of lectures, prepared 
in the interest of teachers and prospective teachers of mathematics in 
German secondary schools. He endeavored to reduce the gap between 
the school and the university, to rouse the schools from the lethargy 
of tradition, to guide the school teaching into directions that would 
stimulate healthy growth; and also to influence university attitude and 
teaching toward a recognition of the normal function of the secondary 
school, to the end that mathematical education should be a continuous 

These lectures of Professor Klein took final form in three printed 
volumes, entitled Elementary Mathematics from an Advanced Standpoint. 

t1a/*A/1 in 


They constitute an invaluable work, serviceable alike to the university 
teacher and to the teacher in the secondary school. There is, at present, 
nothing else comparable with them, either with respect to their skilfully 
integrated material, or to the fascinating way in which this material is 
discussed. This English volume is a translation of Part I of the above 
work. Its preparation is the result of a suggestion made by Professor 
C our ant, of the University of Gottingen. It is the expression of a desire 
to serve the need, in English speaking countries, of actual and prospective 
teachers of mathematics; and it appears with the earnest hope that, in 
a rather free translation, something of the spirit of the original has 
been retained. 

The Translators. 



Introduction 1 

First Part: Arithmetic 

I. Calculating with Natural Numbers 6 

1. Introduction of Numbers in the Schools 6 

2. The Fundamental Laws of Reckoning 8 

3. The Logical Foundations of Operations with Integers 10 

4. Practice in Calculating with Integers 17 

II. The First Extension of the Notion of Number 22 

1. Negative Numbers 23 

2. Fractions 28 

3- Irrational Numbers 31 

III. Concerning Special Properties of Integers 37 

IV. Complex Numbers 55 

1. Ordinary Complex Numbers 55 

2. Higher Complex Numbers, especially Quaternions 58 

3. Quaternion Multiplication Rotation and Expansion 65 

4. Complex Numbers in School Instruction 75 

Concerning the Modern Development and the General Structure of 

Mathematics 77 

Second Part: Algebra 

I. Real Equations with Real Unknowns 87 

1. Equations with one parameter 87 

2. Equations with two parameters 88 

3. Equations with three parameters A, //, v 94 

II. Equations in the field of complex quantities 101 

A. The fundamental theorem of algebra 101 

B. Equations with a complex parameter 104 

1. The "pure" equation HO 

2. The dihedral equation 115 

3. The tetrahedral, the octahedral, and the icosahedral equations . 120 

4. Continuation: Setting up the Normal Equation 124 

5. Concerning the Solution of the Normal Equations 130 

6. Uniformization of the Normal Irrationalities by Means of Trans- 

cendental Functions 133 

7. Solution in Terms of Radicals 138 

8. Reduction of Genral Equations to Normal Equations 141 

Contents. IX 

Third Part: Analysis 

J Page 

I. Logarithmic and Exponential Functions 144 

1. Systematic Account of Algebraic .Analysis 144 

2. The Historical Development of the Theory 146 

3. The Theory of Logarithms in the Schools 155 

4. The Standpoint of Function Theory .156 

II. The Goniometric Functions . . . 162 

1. Theory of the Goniometric Functions 162 

2. Trigonometric Tables 169 

A. Purely Trigonometric Tables 170 

B. Logarithmic Trigonometric Tables 172 

3- Applications of Goniometric Functions 175 

A. Trigonometry, in particular, spherical trigonometry 175 

B. Theory of small oscillations, especially those of the pendulum . 186 

C. Representation of periodic functions by means of series of gonio- 
metric functions (trigonometric series) 190 

III. Concerning Infinitesimal Calculus Proper 207 

1. General Considerations in Infinitesimal Calculus 207 

2. TAYLORS Theorem 223 

3. Historical and Pedagogical Considerations 234 


I. Transcendence of the Numbers e and n 237 

II. The Theory of Assemblages 250 

1. The Power of an Assemblage 251 

2. Arrangement of the Elements of an Assemblage 262 

Index of Names 269 

Index of Contents . 271 


In recent years 1 , a far reaching interest has arisen among university 
teachers of mathematics and natural science directed toward a suitable 
training of candidates for the higher teaching positions. This is really 
quite a new phenomenon. For a long time prior to its appearance, 
university men were concerned exclusively with their sciences, without 
giving a thought to the needs of the schools, without even caring to 
establish a connection with school mathematics. What was the result 
of this practice? The young university student found himself, at the 
outset, confronted with problems which did not suggest, in any particular, 
the things with which he had been concerned at school. Naturally he 
forgot these things quickly and thoroughly. When, after finishing his 
course of study, he became a teacher, he suddenly found himself expected 
to teach the traditional elementary mathematics in the old pedantic 
way; and, since he was scarcely able, unaided, to discern any connection 
between this task and his university mathematics, he soon fell in with 
the time honored way of teaching, and his university studies remained 
only a more or less pleasant memory which had no influence upon his 

There is now a movement to abolish this double discontinuity, helpful 
neither to the school nor to the university. On the one hand, there is 
an effort to impregnate the material which the schools teach with new 
ideas derived from modern developments of science and in accord with 
modern culture. We shall often have occasion to go into this. On the 
other hand, the attempt is made to take into account, in university 
instruction, the needs of the school teacher. And it is precisely in such 
comprehensive lectures as I am about to deliver to you that I see one 
of the most important ways of helping. I shall by no means address 
myself to beginners, but I shall take for granted that you are all ac- 
quainted with the main features of the chief fields of mathematics. I 
shall often talk of problems of algebra, of number theory, of function 
theory, etc., without being able to go into details. You must, therefore, 
be moderately familiar with these fields, in order to follow me. My task 
will always be to show you the mutual connection between problems in 

f 1 Attention is again drawn to the fact that the wording of the text is, almost 
throughout, that of the lithographed volume of 1908 and that comments which 
refer to later years have been put into the appendices.] 

Klein, Elementary Mathematics. 1 

2 Introduction, 

the various fields, a thing which is not brought out sufficiently in the 
usual lecture cours.e, and more especially to emphasize the relation of 
these problems to those of school mathematics. In this way I hope 
to make it easier for you to acquire that ability which I look upon as 
the real goal of your academic study: the ability to draw (in ample 
measure) from the great body of knowledge there put before you a 
living stimulus for your teaching. 

Let me now put before you some documents of recent date which 
give evidence of widespread interest in the training of teachers and 
which contain valuable material for us. Above all I think here of the 
addresses given at the last Meeting of Naturalists held September 16, 
1907, in Dresden, to which body we submitted the Proposals for the 
Scientific Training of Prospective Teachers of Mathematics and Science 
of the Committee on Instruction of the Society of German Naturalists 
and Physicians. You will find these Proposals as the last section in the 
Complete Report of this Committee 1 which, since 1904, has been con- 
sidering the entire complex of questions concerning instruction in mathe- 
matics and natural science and has now ended its activity ; I urge you 
to take notice, not only of these Proposals, but also of the other parts 
of this very interesting report. Shortly after the Dresden meeting there 
occurred a similar debate at the Meeting of German Philologists and 
Schoolmen in Basel, September 25, in which, to be sure, the mathematical- 
scientific reform movement was discussed only as a link in the chain 
of parallel movements occurring in philological circles. After a report 
by me concerning our aims in mathematical-natural science reform there 
were addresses by P. Wendland (Breslau) on questions in Archeology, 
Al. Brandl (Berlin) on modern languages and , finally , Ad. Harnack (Berlin) 
on History and religion. These four addresses appeared together in one 
broschure 2 to which I particulary refer you. I hope that this auspicious 
beginning will develop into further cooperation between our scientists 
and the philologists, since it will bring about friendly feeling and mutual 
understanding between two groups whose relations have been unsympa- 
thetic even if not hostile. Let us endeavor always to foster such good 
relations even if we do among ourselves occasionally drop a critical 
word about the philologists, just as they may about us. Bear in mind 
that you will later be called upon in the schools to work together with 
the philologists for the common good and that this requires mutual 
understanding and appreciation. 

Along with this evidence of efforts which reach beyond the borders 
of our field, I should like to mention a few books which aim in the 

1 Die Tdtigkeit der Unterrichtskommission der Gesellschajt deutscher Natur- 
forscher und Arzte, edited by A. Gutzmer. Leipzig and Berlin, 1908. 

2 Universitdt und Schule. Addresses delivered by F.Klein, P. Wendland, 
Al. Brandl, Ad. Harnack. Leipzig 1907. 

Introduction. 3 

same direction in the mathematical field and which will therefore be 
important for these lectures. Three years ago I gave, for the first time, 
a course of lectures with a similar purpose. My assistant at that time, 
R. Schimmack, worked the material up and the first part has recently 
appeared in print 1 . In it are considered the different kinds of schools, 
including the university, the conduct of mathematical instruction in 
them, the interests that link them together, and other similar matters. 
In what follows I shall from time to time refer to things which appear 
there without repeating them. This makes it possible for me to extend 
somewhat those considerations. That volume concerns itself with the 
organization of school instruction. I shall now consider the mathematical 
content of the material which enters into that instruction. If I frequently 
advert to the actual conduct of instruction in the schools, my remarks 
will be based not merely upon indefinite pictures of how the thing 
might be done or even upon dim recollections of my own school days; 
for I am constantly in touch with Schimmack, who is now teaching in 
the Gottingen gymnasium and who keeps me informed as to the present 
state of instruction, which has, in fact, advanced substantially beyond 
what it was in earlier years. During this winter semester I shall discuss 
"the three great AV, that is arithmetic, algebra, and analysis, with- 
holding geometry for a continuation of the course during the coming 
summer. Let me remind you that, in the language of the secondary 
schools, these three subjects are classed together as arithmetic, and 
that we shall often note deviations in the terminology of the schools as 
compared with that at the universities. You see, from this small illustra- 
tion, that only living contact can bring about understanding. 

As a second reference I shall mention the three volume Enzyklopadie 
der Elementarmathematik by H. Weber and J. Wellstein, the work which, 
among recent publications, most nearly accords with my own tendencies. 
For this semester, the first volume, Enzyklopadie der elementaren Algebra 
und Analysis, prepared by H. Weber 2 , will be the most important. I 
shall indicate at once certain striking differences between this work and 
the plan of my lectures. In Weber- Wellstein, the entire structure of 
elementary mathematics is built up systematically and logically in the 
mature language of the advanced student. No account is taken of how 
these things actually may come up in school instruction. The present- 
ation in the schools, however, should be psychological and not syste- 
matic. The teacher so to speak, must be a diplomat. He must take 
account of the psychic processes in the boy in order to grip his interest ; 

1 Klein, F., Vortrage uber den mathematischen Unterricht an hoheren Schulen. 
Prepared by von R. Schimmack. Part 1. Von der Organisation des mathematischen 
Unterrichts. Leipzig 1907. This book is referred to later as "Klein-Schimmack". 

2 Second edition. Leipzig 1906. [Fourth edition, 1922, revised by P. Epstein. 
Referred to as "Weber- Wellstein I". 

4 Introduction. 

and he will succeed only if he presents things in a form intuitively 
comprehensible. A more abstract presentation will be possible only in 
the upper classes. For example: The child cannot possibly understand 
if numbers are explained axiomatically as abstract things devoid of 
content, with which one can operate according to formal rules. On the 
contrary, he associates numbers with concrete images. They are numbers 
of nuts, apples, and other good things, and in the beginning they can 
be and should be put before him only in such tangible form. While this 
goes without saying, one should mutatis mutandis take it to heart, 
that in all instruction, even in the university, mathematics should be 
associated with everything that is seriously interesting to the pupil at 
that particular stage of his development and that can in any way be 
brought into relation with mathematics. It is just this which is back 
of the recent efforts to give prominence to applied mathematics at the 
university. This need has never been overlooked in the schools so much 
as it has at the university. It is just this psychological value which I 
shall try to emphasize especially in my lectures. 

Another difference between Weber- Wellstein and myself has to do 
with defining the content of school mathematics. Weber and Wellstein 
are disposed to be conservative, while I am progressive. These things 
are thoroughly discussed in Klein-Schimmack. We, who are called the 
reformers, would put the function concept at the very center of in- 
struction, because, of all the concepts of the mathematics of the past 
two centuries, this one plays the leading role wherever mathematical 
thought is used. We would introduce it into instruction as early as 
possible with constant use of the graphical method, the representation 
of functional relations in the xy system, which is used today as a matter 
of course in every practical application of mathematics. In order to 
make this innovation possible, we would abolish much of the traditional 
material of instruction, material which may in itself be interesting, but 
which is less essential from the standpoint of its significance in con- 
nection with modern culture. Strong development of space perception, 
above all, will always be a prime consideration. In its upper reaches, 
however, instruction should press far enough into the elements of in- 
finitesimal calculus for the natural scientist or insurance specialist to 
get at school the tools which will be indispensable to him. As opposed 
to these comparatively recent ideas, Weber- Wellstein adheres essentially 
to the traditional limitations as to material. In these lectures I shall of 
course be a protagonist of the new conception. 

My third reference will be to a very stimulating book: Didaktik und 
Methodik des Rechnens und der Mathematik 1 by Max Simon, who like 

1 Second edition, Munich 1908. Separate reprint from Baumeister's Hand- 
buck der Erziehungs- und Unterrichtslehre filr hohere Schulen, first edition, 1895. 

Introduction. 5 

Weber and Wellstein is at Strassburg. Simon is often in agrement 
with our views, but he sometimes takes the opposite standpoint ; and 
inasmuch as he is a very subjective, temperamental, personality he often 
clothes these contrasting views in vivid words. To give one example, 
the proposals of the Committee on Instruction of the Natural Scientists 
require an hour of geometric propaedeutics in the second year of the 
gymnasium, whereas at the present time this usually begins in the third 
year. It has long been a matter of discussion which plan is the better; 
and the custom in the schools has often changed. But Simon declares 
the position taken by the Commission, which, mind you, is at worst 
open to argument, to be ''worse than a crime", and that without in the 
least substantiating his judgment. One could find many passages of 
this sort. As a precursor of this book I might mention Simon's Methodik 
der elementaren Arithmetik in Verbindung mit algebraischer Analysis 1 . 
After this brief introduction let us go over to the subject proper, 
which I shall consider under three headings, as above indicated. 

Leipzig 1906. 

First Part 


I. Calculating with Natural Numbers 

We begin with the foundation of all arithmetic, calculation with 
positive integers. Here, as always in the course of these lectures, we 
first raise the question as to how these things are handled in the schools ; 
then we shall proceed to the question as to what they imply when 
viewed from an advanced standpoint. 

1. Introduction of Numbers in the Schools 

I shall confine myself to brief suggestions. These will enable you 
to recall how you yourselves learned your numbers. In such an exposi- 
tion it is, of course, not my purpose to induct you into the practice 
of teaching, as is done in the Seminars of the secondary schools. I shall 
merely exhibit the material upon which we shall base our critique. 

The problem of teaching children the properties of integers and how 
to reckon with them, and of leading them on to complete mastery, is 
very difficult and requires the labor of several years, from the first school 
year until the child is ten or eleven years old. The manner of instruction 
as it is carried on in this field in Germany can perhaps best be designated 
by the words intuitive and genetic, i. e., the entire structure is gradually 
erected on the basis of familiar, concrete things, in marked contrast to 
the customary logical and systematic method at the university. 

One can divide up this material of instruction roughly as follows: 
The entire first year is occupied with the integers from 1 to 20, the 
first half being devoted to the range 1 to 10. The integers appear at 
first as numbered pictures of points or as arrays of all sorts of objects 
familiar to the children. Addition and multiplication are then presented 
by intuitional methods, and are fixed in mind. 

In the second stage, the integers from 1 to 100 are considered and the 
Arabic numerals, together with the notion of positional value and the 
decimal system, are introduced. Let us note, incidentally, that the name 
"Arabic numerals", like so many others in science, is a misnomer. 
This form of writing was invented by the Hindus, not by the Arabs. 
Another principal aim of the second stage is knowledge of the multi- 

Introduction of Numbers in the Schools. 7 

plication table. One must know what 5 X 7 or 3 x 8 is in one's sleep, 
so to speak. Consequently the pupil must learn the multiplication table 
by heart to this degree of thoroughness, to be sure only after it has 
been made clear to him visually with concrete things. To this end the 
abacus is used to advantage. It consists, as you all know, of 10 wires 
stretched one above another, upon each of which there are strung ten 
movable beads. By sliding these beads in the proper way, one can 
read off the result of multiplication and also its decimal form. 

The third stage, finally, teaches calculation with numbers of more 
than one digit, based on the known simple rules whose general validity 
is evident, or should be evident, to the pupil. To be sure, this evidence 
does not always enable the pupil to make the rules completely his own ; 
they are often instilled with the authoritative dictum: "It is thus and 
so, and if you don't know it yet, so much the worse for you!" 

I should like to emphasize another point in this instruction which is 
usually neglected in university teaching. It is that the application of 
numbers to practical life is strongly emphasized. From the beginning, 
the pupil is dealing with numbers taken from real situations, with coins, 
measures, and weights; and the question, "What does it cost ?", which is so 
important in daily life, forms the pivot of much of the material of instruc- 
tion. This plan rises soon to the stage of problems, when deliberate 
thought is necessary in order to determine what calculation is demanded. 
It leads to the problems in proportion, alligation, etc. To the words 
intuitive and genetic, which we used above to designate the character 
of this instruction, we can add a third word, applications. 

We might summarize the purpose of the number work by saying: 
It aims at reliability in the use of the rules of operation, based on a parallel 
development of the intellectual abilities involved, and without special concern 
for logical relations. 

Incidentally, I should like to direct your attention to a contrast 
which often plays a mischievous role in the schools, viz., the contrast 
between the university-trained teachers and those who have attended - 
normal schools for the preparation of elementary school teachers. The 
former displace the latter, as teachers of arithmetic, during or after 
the sixth school year, with the result that a regrettable discontinuity 
often manifests itself. The poor youngsters must suddenly make the 
acquaintance of new expressions, whereas the old ones are forbidden. 
A simple example is the different multiplication signs, the x being pre- 
ferred by the elementary teacher, the point by the one who has attended 
the university. Such conflicts can be dispelled, if the more highly 
trained teacher will give more heed to his colleague and will try to meet 
him on common ground. That will become easier for you, if you will 
realize what high regard one must have for the performance of the 
elementary school teachers. Imagine what methodical training is ne- 

g Arithmetic: Calculating with Natural Numbers. 

cessary to indoctrinate over and over again a hundred thousand stupid, 
unprepared children with the principles of arithmetic ! Try it with your 
university training; you will not have great success! 

Returning, after this digression, to the material of instruction, we 
note that after the third year of the gymnasium*, and especially in the 
fourth year, arithmetic begins to take on the more aristocratic dress of 
mathematics, for which the transition to operations with letters is charac- 
teristic. One designates by a, b, c, or x, y 3 z any numbers, at first only 
positive integers, and applies the rules and operations of arithmetic to 
the numbers thus symbolized by letters, whereby the numbers are 
devoid of concrete intuitive content. This represents such a long step 
in abstraction that one may well declare that real mathematics begins 
with operations with letters. Naturally this transition must not be 
accomplished rapidly. The pupils must accustom themselves gradually 
to such marked abstraction. 

It seems unquestionably necessary that, for this instruction, the 
teacher should know thoroughly the logical laws and foundations of 
reckoning and of the theory of integers. 

2. The Fundamental Laws of Reckoning 

Addition and multiplication were familiar operations long before 
any one inquired as to the fundamental laws governing these operations. 
It was in the twenties and thirties of the last century that particularly 
English and French mathematicians formulated the fundamental pro- 
perties of the operations, but I will not enter into historical details here. 
If you wish to study these, I recommend to you, as I shall often do, 
the great Enzyklopddie der Mathematischen Wissenschaften mil Ein- 
schlufi Hirer Anwendungen 1 , and also the French translation: Encyclope- 
dic des Sciences mathematiques pures et appliquees 2 which bears in part 
the character of a revised and enlarged edition. If a school library 
has only one mathematical work, it ought to be this encyclo- 
pedia, for through it the teacher of mathematics would be placed in 
position to continue his work in any direction that might interest him. 
For us, at this place, the article of interest is the first one in the first 
volume 3 H.Schubert: "Grundlagen der Arithmetik", of which the trans- 
lation into French is by Jules Tannery and Jules Molk. 

* The German gymnasium is a nine-year secondary school, following a four- 
year preparatory school. Hence the third year of the gymnasium is the student's 
seventh school year. 

1 Leipzig (B. G. Teubner) from 1908 on. Volume I has appeared complete, 
Volumes II VI are nearing completion. 

2 Paris (Gauthur-Villars) and Leipzig (Teubner) from 1904 on; unfortunately 
the undertaking had to be abandoned after the death of its editor J. Molk (1914). 

8 Arithmetik und Algebra, edited by W. Fr. Meyer (1896 1904) ; in the French 
edition, the editor was J. Molk. 

The Fundamental Laws of Reckoning. g 

Going back to our theme, I shall enumerate the five fundamental 
laws upon which addition depends: 

1. a + b is always again a number, i. e., addition is always possible 
(in contrast to subtraction, which is not always possible in the domain 
of positive integers). 

2. a + b is one-valued. 

3. The associative law holds: 

(a + b) + c = a+(b + c), 

so that one may omit the parentheses entirely. 

4. The commutative law holds: 

a + b = b + a . 

5. The monotonic law holds: 

If b > c , then a + b > a + c . 

These properties are all obvious immediately if one recalls the process 
of counting; but they must be formally stated in order to justify logically 
the later developments. 

For multiplication there are five exactly analogous laws: 

\. a b is always a number. 

2. a b is one-valued. 

3. Associative law: a (b c) = (a b) c = a b c. 

4. Commutative law: a b = b a. 

5. Monotonic law: If b > c, then a b > a c. 

Multiplication together with addition obeys also the following law. 

6. Distributive law: 

a-(b + c)=a'b + a-c. 

It is easy to show that all elementary reckoning can be based upon 
these eleven laws. It will be sufficient to illustrate this fact by a simple 
example, say the multiplication of 7 and 12. From the distributive 
law we have: 

7-12 = 7- (10 + 2) = 70 + 14, 

and if we separate 14 into 10 + 4 (carrying the tens), we have, by the 
associative law of addition, 

70 + (10 + 4) = (70 + 10) + 4 = 80 + 4 = 84. 

You will recognize in this procedure the steps of the usual decimal 
reckoning. It would be well for you to construct for yourselves more 
complicated examples. We might summarize by saying that ordinary 
reckoning with integers consists in repeated use of the eleven fundamental 
laws together with the memorized results of the addition and multiplication 

But where does one use the monotonic laws? In ordinary formal 
reckoning, to be sure, they are superfluous, but not in certain other 

JO Arithmetic: Calculating with Natural Numbers. 

problems. Let me remind you of the process called abridged multiplication 
and division with decimal numbers 1 . That is a thing of great practical 
importance which unfortunately is too little known in the schools, as 
well as among university students, although it is sometimes mentioned 
in the second year of the gymnasium. As an example, suppose that 
one wished to compute 567 134, and that the units digit in each number 
was of questionable accuracy, say as a result of physical measurement. 
It would be unnecessary work, then, to determine the product exactly, 
since one could not guarantee an exact result. It is, however, important 
to know the order of magnitude of the product, i. e., to know between 
which tens or between which hundreds the exact value lies. The mono- 
tonic law supplies this estimate at once; for it follows by that law that 
the desired value lies between 560-134 and 570 134 or between 560-130 
and 570-140. I leave to you the carrying out of the details; at least 
you see that the monotonic law is continually used in abridged reckoning. 
A systematic exposition of these fundamental laws is, of course, not 
to be thought of in the secondary schools. After the pupils have gained 
a concrete understanding and a secure mastery of reckoning with 
numbers, and are ready for the transition to operations with letters, 
the teacher should take the opportunity to state, at least, the associative, 
commutative, and distributive laws and to illustrate them by means 
of numerous obvious numerical examples. 

3. The Logical Foundations of Operations with Integers 

While instruction in the schools will naturally not rise to still more 
difficult questions, present mathematical investigation really begins with 
the question : How does one justify the above-mentioned fundamental laws, 
how does one account for the notion of number at all? I shall try to explain 
this matter in accordance with the announced purpose of these lectures 
to endeavor to get new light upon school topics by looking at them from 
another point of view. I am all the more willing to do this because 
these modern thoughts crowd in upon you from all sides during your 
academic years, but not always accompanied by any indication of their 
psychological significance. 

First of all, so far as the notion of number is concerned, it is very 
difficult to discover its origin. Perhaps one is happiest if one decides 
to ignore these most difficult things. For more complete information 
as to these questions, which are so earnestly discussed by the philo- 
sophers, I must refer you to the article, already mentioned, in the 
French encyclopedia, and I shall confine myself to a few remarks. A 
widely accepted belief is that the notion of number is closely connected 
with the notion of time, with temporal succession. The philosopher Kant 

1 The monotonic laws will be used later, also, in the theory of irrational numbers. 

The Logical Foundations of Operations with Integers. \\ 

and the mathematician Hamilton represent this view. Others think 
that number has more to do with space perception. They base the 
notion of number upon the simultaneous perception of different objects 
which are near each other. Still others see, in number concepts, the 
expression of a peculiar faculty of the mind which exists independently 
of, and coordinate with, or even above, perception of space and time. 
I think that this conception would be well characterized by quoting 
from Faust the lines which Minkowski, in the preface of his book on 
Diophantine Approximation, applies to numbers: 

"Gottinnen thronen hehr in Einsamkeit, 
Um sie kein Ort, noch weniger eine Zeit." 

While this problem involves primarily questions of psychology and 
epistemology, the justification of our eleven laws, at least the recent 
researches regarding their compatibility, implies questions of logic. We 
shall distinguish the following four points of view. 

1. According to the first of these, best represented perhaps by Kant, 
the rules of reckoning are immediate necessary results of perception, 
whereby this word is to be understood, in its broadest sense, as "inner 
perception 1 ' or intuition. It is not to be understood by this that mathe- 
matics rests throughout upon experimentally controllable facts of ex- 
ternal experience. To mention a simple example, the commutative law 
is established by examining the accompanying picture, which 
consists of two rows of three points each, that is, 2 3 = 3 2. If 

the objection is raised that in the case of only moderately large 
numbers, this immediate perception would not suffice, the reply is that 
we call to our assistance the theorem of mathematical induction. If a 
theorem holds for small numbers, and if an assumption of its validity for 
a number n always insures its validity forn-{-\, then it holds generally 
for every number. This theorem, which I consider to be really an in- 
tuitive truth, carries us over the boundary where sense perception fails. 
This standpoint is more or less that of Poincare in his well known 
philosophical writings. 

If we would realize the significance of this question as to the source 
of the validity of our eleven fundamental rules of reckoning, we should 
remember that, along with arithmetic, mathematics as a whole rests 
ultimately upon them. Thus it is not asserting too much to say, that, 
according to the conception of the rules of reckoning which we have 
just outlined, the security of the entire structure of mathematics rests upon 
intuition, where this word is to be understood in its most general sense. 

2. The second point of view is a modification of the first. According 
to it, one tries to separate the eleven fundamental laws into a larger 
number of shorter steps of which one need take only the simplest 
directly from intuition, while the remainder are deduced from these 
by rules of logic without any further use of intuition. Whereas, before, 

\2 Arithmetic: Calculating with Natural Numbers. 

the possibility of logical operation began after the eleven fundamental 
laws had been set up, it can start earlier here, after the simpler ones 
have been selected. The boundary between intuition and logic is displaced 
in favor of the latter. Hermann Grassmann did pioneer work in this 
direction in his Lehrbuch der Arithmetik 1 in 1861. As an example from 
it, I mention merely that the commutative law can be derived from 
the associative law by the aid of the principle of mathematical induction. 
Because of the precision of his presentation, one might place by the 
side of this book of Grassmann one by the Italian Peano, Arithmetices 
principia nova methodo exposita*. Do not assume, however, because of 
this title, that the book was written in Latin ! It is written in a peculiar 
symbolic language designed by the author to display each logical step 
of the proof and emphasize it as such. Peano wishes to have a guarantee 
in this way, that he is making use only of the principle which he specifi- 
cally mentions, with nothing whatever coming from intuition. He wishes 
to avoid the danger that countless uncontrollable associations of ideas 
and reminders of perception might creep in if he used our ordinary 
language. Note, too, that Peano is the leader of an extensive Italian 
school which is trying in a similar way to separate into small groups 
the premises of each individual branch of mathematics, and, with the 
aid of such a symbolic language, to investigate their exact logical 

3. We come now to a modern extension of these ideas, which has, 
moreover, been influenced by Peano. I refer to that treatment of the 
foundations of arithmetic which puts the theory of point sets into the 
foreground. You will be able to form a notion of the wide range 
of the idea of a point set if I tell you that the totality of all integers, 
as well as that of all points on a line segment, are special examples 
of point sets. Georg Cantor, as is generally known, was the first 
to make this general idea the object of orderly mathematical 
speculation. The theory of point sets, which he created, is now 
claiming the profound attention of the younger generation of 
mathematicians. Later I shall endeavor to give you a cursory view 
of this subject. For the present, it is sufficient to characterize as follows 
the tendency of the new foundation of arithmetic which have been based 
upon it: The properties of integers and of operations with them are to 
be deduced from the general properties and abstract relations of point sets, 
in order that the foundation may be as sound and general as possible. 

1 With the addition to the title "fur hohere Lehranstalten" (Berlin 1861). 
The corresponding chapters are reprinted in H. Grassmann f s Gesammelten mathe- 
matischen und physikalischen Werken (edited by F. Engel), Vol. II, 1, pp. 295 349- 
Leipzig 1904. 

2 Augustae Taurinorum. Torino 1889. [There is a more comprehensive 
presentation in Peano's Formulaire de MatMmatiques (18921899). 

The Logical Foundations of Operations with Integers. ja 

One of the pioneers along this path was Richard Dedekind, who, in his 
small but important book Was sind und was sollen die Zahlen? *, attempted 
such a foundation for integers. H. Weber inclines to this point of view 
in the first part of Weber-Wellstein, volume I (See p. 3). To be sure, 
the deduction is quite abstract and offers, still, certain grave difficulties, 
so that Weber, in an Appendix to Volume III 2 , gave a more elementary 
presentation, using only finite point sets. In later editions, this appendix 
is incorporated into Volume I. Those of you who are interested in such 
questions are especially referred to this presentation. 

4. Finally, I shall mention the purely formal theory of numbers, which, 
indeed, goes back to Leibniz and which has recently been brought into 
the foreground again by Hilbert. His address Vber die Grundlagen der 
Logik und Arithmetik* at the Heidelberg Congress in 1904 is important 
for arithemtic 3 . His fundamental conception is as follows: Once one 
has the eleven fundamental rules of reckoning, one can operate with the 
letters a, b, c t . . ., which actually represent arbitrary integers, without 
bearing in mind that they have a real meaning as numbers. In other 
words: let a, b, c, . . . , be things devoid of meaning, or things of whose 
meaning we know nothing; let us agree only that one may combine 
them according to those eleven rules, but that these combinations need 
not have any real known meaning. Obviously one can than operate 
with a,b,c, . . ., precisely as one ordinarily does with actual numbers. 
Only the question arises here whether these operations could lead one to 
contradictions. Now ordinarily one says that intuition shows us the 
existence of numbers for which these eleven laws hold, and that it is 
consequently impossible for contradictions to lurk in these laws. But 
in the present case, where we are not thinking of the symbols as having 
definite meaning, such an appeal to perception is not permissible. In 
fact, there arises the entirely new problem, to prove logically that no oper- 
ations with our symbols which are based on the eleven fundamental laws 
can ever lead to a contradiction, i. e., that these eleven laws are consistent, 
or compatible. While we were discussing the first point of view, we took 
the position that the certainty of mathematics rests upon the existence 
of intuitional things which fit its theorems. The adherents of this formal 
standpoint, on the other hand, must hold that the certainty of mathematics 
rests upon the possibility of showing that the fundamental laws considered 
formally and without reference to their intuitional content, constitute a 
.logically consistent system. 

1 Braunschweig 1888; third edition 1911. 

2 Angewandte Elementarmathematik. Revised by H. Weber, J. Wells tein, 
R. H.Weber. Leipzig 1907- 

* On the foundations of logic and arithmetic. 

3 Verhandlungen des 3. international Mathematikerkongr esses in Heidelberg 
August 813, 1904, p. 174 et seq., Leipzig 1905. 

{4 Arithmetic: Calculating with Natural Numbers. 

I shall close this discussion with the following remarks: 

a) Hilbert indicated all of these points of view in his Heidel- 
berg address, but he followed none of them through completely. 
Afterwards he pushed them somewhat farther in a course of lectures, 
but then abandoned them. We can thus say that here is a field for 
investigation 1 . 

b) The tendency to crowd intuition completely off the field and to 
attain to really pure logical investigations seems to me not completely 
feasible. It seems to me that one must retain something, albeit a minimum, 
of intuition. One must always use a certain intuition in the most ab- 
stract formulation with the symbols one uses in operations, in order to 
recognize the symbols again, even if one thinks only about the shape of 
the letters. 

c) Let us even assume that the proposed problem has been solved 
in a way free from objection, that the compatibility of the eleven funda- 
mental laws has been proved logically. Precisely at this point an opening 
is offered for a remark which I should like to make with the utmost 
emphasis. One must see clearly that the real arithmetic, the theory of actual 
integers, is neither established, nor can ever be established, by considerations 
of this nature. It is impossible to show in a purely logical way that the 
laws whose consistency is established in that manner are actually valid 
for the numbers with which we are intuitionally familiar; that the 
undefined things of which we speak, and the operations which we apply 
to them, can be identified with actual numbers and with the processes 
of addition and multiplication in their intuitively clear significance. 
What is accomplished is, rather, that the tremendous problem of building 
the foundations of arithmetic, unassailable in its complexity, is split into 
two parts, and that the first, the purely logical problem, the setting up 
of independent fundamental laws or axioms and the investigation of them 
as to independence and consistency has been made available to study. 
The second, the more epistemological part of the problem, which has 
to do with the justification for the application of these laws to actual 
conditions, is not even touched, although it must of course be solved 
also if one will really build the foundations of arithmetic. This second 
part presents, in itself, an extremely profound problem, whose diffi- 
culties lie in the general field of epistemology. I can characterize its 
standing most clearly perhaps, by the somewhat paradoxical remark 
that anyone who tolerates only pure logic in investigations in pure 
mathematics must, to be consistent, look upon the second part of the 
problem of the foundation of arithmetic, and hence upon arithmetic 
itself, as belonging to applied mathematics. 

P Concerning more recent developments in these investigations, see the pre- 
ceding footnote.] 

The Logical Foundations of Operations with Integers. \ 5 

I have felt obliged to go into detail here very carefully, in as much 
as misunderstandings occur so often at this point, because people simply 
overlook the existence of the second problem. This is by no means the 
case with Hilbert himself, and neither my disagreement nor my agree- 
ment with him is a warranted conclusion if it be based on such an 

Thomae of Jena, coined the neat expression "thoughtless thinkers" 
for those persons who confine themselves exclusively to these abstract 
investigations concerning things that are devoid of meaning, and to 
theorems that tell nothing, and who forget not only that second problem 
but often also all the rest of mathematics. This facetious term cannot 
apply, of course, to people who carry on those investigations alongside 
of many others of a different sort. 

In connection with this brief survey of the foundation of arithmetic, 
I shall bring to your notice a few general matters. Many have thought 
that one could, or that one indeed must, teach all mathematics deduc- 
tively throughout, by starting with a definite number of axioms and de- 
ducing everything from these by means of logic. This method, which 
some seek to maintain upon the authority of Euclid, certainly does not 
correspond to the historical development of mathematics. In fact, 
mathematics has grown like a tree, which does not start at its tiniest 
rootlets and grow merely upward, but rather sends its roots deeper 
and deeper at the same time and rate that its branches and leaves are 
spreading upward. Just so if we may drop the figure of speech , mathe- 
matics began its development from a certain standpoint corresponding 
to normal human understanding, and has progressed, from that point, 
according to the demands of science itself and of the then prevailing 
interests, now in the one direction toward new knowledge, now in the 
other through the study of fundamental principles. For example, our 
standpoint today with regard to foundations is different from that of 
the investigators of a few decades ago ; and what we today would state 
as ultimate principles, will certainly be outstripped after a time, in 
that the latest truths will be still more meticulously analyzed and 
referred back to something still more general. We see, then, that as 
regards the fundamental investigations in mathematics, there is no final 
ending, and therefore, on the other hand, no first beginning, which could 
offer an absolute basis for instruction. 

Still another remark concerning the relation between the logical and 
the intuitional handling of mathematics, between pure and applied 
mathematics. I have already emphasized the fact that, in the schools, 
applications accompany arithmetic from the beginning, that the pupil 
learns not only to understand the rules, but to do something with them. 
And it should always be so in the teaching of mathematics ! Of course, 
the logical connections, one might say the rigid skeleton in the mathematical 

16 Arithmetic: Calculating with Natural Numbers. 

organism, must remain, in order to give it its peculiar trustworthiness. 
But the living thing in mathematics, its most important stimulus, its 
effectiveness in all directions, depends entirely upon the applications, 
i. e., upon the mutual relations between those purely logical things and 
all other domains. To banish applications from mathematics would be 
comparable to seeking the essence of the living animal in the skeleton 
alone, without considering muscles, nerves and tissues, instincts, in short, 
the very life of the animal. 

In scientific investigation there is often, to be sure, a division of labor 
between pure and applied science, but when this happens, provision 
must be made otherwise for maintaining their connection if conditions 
are to remain sound. In any case, and this should be especially emphasiz- 
ed here, for the school such a division of labor, such a fareaching specializ- 
ation of the individual teacher, is not possible. To put the matter crassly, 
imagine that at a certain school a teacher is appointed who treats 
numbers only as meaningless symbols, a second teacher who knows how 
to bridge the gap frdm these empty symbols to actual numbers, a third, 
a fourth,. a fifth, finally, who understands the application of these 
numbers to geometry, to mechanics, and to physics; and that these 
different teachers are all turned lose upon the pupils. You see that 
such an organization of teaching is impossible. In this way, the things 
could not be brought to the comprehension of the pupils, neither would 
the individual teachers be able even to understand each other. The 
needs of school instruction itself require precisely a certain many sided- 
ness of the individual teacher, a comprehensive orientation in the field 
of pure and applied mathematics, in the broadest sense, and in- 
clude thus a desirable remedy against a too extensive splitting up of 

In order to give a practical turn to the last remarks I refer again to 
our above mentioned Dresden Proposals. There we recommend outright 
that applied mathematics, which since 1898 has been a special subject 
in the examination for prospective teachers, be made a required part 
in all normal mathematical training, so that competence to teach pure 
and applied mathematics should always be combined. In addition to 
this, it should be noted that, in the Meran Curriculum 1 of the Commis- 
sion of Instruction, the following three tasks are announced as the 
purpose of mathematical instruction in the last school year: 

\. A scientific survey of the systematic structure of mathematics. 

2. A certain degree of skill in the complete handling, numerical and 
graphical, of problems. 

1 Reformvorschl&ge fur den mathematischen und naturwissenschaftlichen Unter- 
richt, iiberreicht der Versammlung der Naturforscher und Arzte zu Meran. Leipzig, 
1905. See also a reprint in the Gesamtbericht der Kommission, p. 93, as well as 
in Klein-Schimmack, p. 208. 

Practice in Calculating with Integers. \j 

3. An appreciation of the significance of mathematical thought for a 
knowledge of nature and for modern culture. 

All these formulations I approve with deep conviction. 

4. Practice in Calculating with Integers 

Turning from discussions which have been chiefly abstract, let us 
give our attention to more concrete things by considering the carrying 
out of numerical calculation. As suitable literature for collateral reading, 
I should mention first of all, the article on Numerisches Rechnen by 
R. Mehnicke 1 in the Enzyclopadie. I can best give you a general view 
of the things that belong here by giving a brief account of this article. 
It is divided into two parts: A. Die Lehre vom genauen Rechnen*, and 
B. Die Lehre vom gendherten Rechnen**. Under A occur all methods 
for simplifying exact calculation with large integers. Convenient devices 
for calculating, tables of products and squares, and in particular, calcu- 
lating machines, which we shall discuss soon. Under B, on the other 
hand, one finds a discussion of the methods and devices for all calculating 
in which only the order of magnitude of the result is important, especially 
logarithmic tables and allied devices, the slide rule, which is only an 
expecially well-arranged graphical logarithmic table; finally, also, the 
numerous important graphical methods. In addition to this reference I 
can recommend the little book by J. Liiroth, Vorlesungen uber nume- 
risches Rechnen****, which, written in agreeable form by a master of the 
subject, gives a rapid survey of this field. 

From the many topics that have to do with calculating with integers, 
I shall select for discussion only the calculating machine, which you will 
find in use, in a great variety of ingenious forms, by the larger banks 
and business houses, and which is really of the greatest practical signi- 
ficance. We have in our mathematical collection one of the most widely 
used types, the "Brunsviga", manufactured by the firm Brunsviga- 
Maschinenwerke Grimme, Natalis & Co. A.-G. in Braunschweig. The design 
originated with the Swedish engineer Odhner, but it has been much chan- 
ged and improved. Is hall describe the machine here in some detail, as a 
typical example. You will find other kinds described in the books mentioned 
above 3 . My description of course can give you a real understanding of the 

1 Enzyklopadie der mathematischen Wissenschaften, Band I, Teil II. See 
also v. Sanden, H., Practical Mathematical Analysis (Translation by Levy), Button 
& Co. Horsburgh, E. M., Modern Instruments and Methods of Calculation. 
Bell & Sons. 

* The Theory of Exact Calculation. 
** The Theory of Approximate Calculation. 

2 Leipzig 1900. 

*** Lectures on Numerical Calculation. 

[ 3 Concerning other types of calculating machines, see also A. Galle, Mathe- 
matische Instrumente, Leipzig 1912.] 

Klein. Elementary Mathematics. 2 

Arithmetic: Calculating with Natural Numbers. 

machine only if you examine it afterwards personally and if you see, 
by actual use, how it is operated. The machine will be at your disposal, 
for that purpose, after the lecture. 

So far as the external appearance of the Brunsviga is concerned, it 
presents schematically a picture somewhat as follows (see Fig. 1, p. 18). 
There is a fixed frame, the "drum", below which and sliding on it, is 
a smaller longish case, the "slide". A handle which projects from the 
drum on the right, ^is operated by hand. On the drum there is a series 
of parallel slits, each of which carries the digits 0,1, 2,. ..,9, read 
downwards; a peg s projects from each slit and can be set at pleasure 
at any one of the ten digits. Corresponding to each of these slits there 
is an opening on the slide under which a digit can appear. Figure 3, p. 19 
gives a view of a newer model of the machine. 

I think that the arrangement of the machine will be clearer if I 
describe to you the process of carrying out a definite calculation, and 

the way in which the machine 
brings it about. For this I select 

The procedure is as follows: 
One first sets the drum pegs on the 
multiplicand, i. e., beginning at 
the rigftt, one puts the first 
lever at the one's digit, the se- 
cond at the ten's digit of the 
multiplicand, etc. If, for example, 
the multiplicand is 12, one sets 
the first lever at 2, the second 
lever at 1 ; all the other levers 
remain at zero (see Fig. 1).. Now 
turn the handle once around, 
clockwise. The multiplicand ap- 
pears under the openings of the slide, in our case a 2 in the first opening 
from the right, a 1 in the second, while zeros remain in all the others. 
Simultaneously, however, in the first of a series of openings in the slide, at 
the left, the digit 1 appears to indicate that we have turned the handle 
once (Fig. 2). // now one has to do with a. multiplier of one digit, one turns 
the handle as many times as this digit indicates; the multiplier will then 
be exhibited on the slide to the left, while the product will appear on the 
slide to the right. How does the apparatus bring this result about ? In 
the first place there is attached to the under side of the slide, at the 
left, a cogwheel which carries, equally spaced on its rim, the digits 
0, 1 , 2, . . . , 9- By means of a driver, this cogwheel is rotated through 
one tenth of its perimeter with every turn of the handle, so that a digit 
becomes visible through the opening in the slide, which actually indicates 


























' ' 

' ' 

' ' 


Fig. i. Before the first turn. 





















I i 









Fig. 2. After the first, turn. 

Practice in Calculating with Integers. 


Fig. 3. 

the number of revolutions, in other words the multiplier. Now as to 
the obtaining of the product, it is brought about by similar cogwheels, 
one under each opening at the right of the slide. But how is it that 
by one and the same turning of the handle, one of these wheels, in the 
above case, moves by 
one unit, the other 
by two? This is 
where the peculiarity 
in construction of the 
Brunsviga appears. 
Under each slit of the 
drum there is a flat 
wheel-shaped disc 
(driver) attached to 
the axle of the handle, 
upon which there are 
nine teeth which are 
movable in a radial di- 
rection (see Fig. 4). By 
means of the projecting 
peg 5 , mentioned above, 
one can turn a ringf/?' 
which rests upon the 
periphery of the disc, 
so that, according to 
the mark upon which 
one sets S in the slit, 
0, 1, 2, . . ., 9 of the 
movable teeth spring outward (in 
Fig. 4, two teeth). These teeth 
engage the cogs under the corre- 
sponding openings of the slide, so 
that with one turn of the handle 
each driver thrusts forward the corre- 
sponding cogwheel by as many units 
as there are teeth pushed out, i.e., by 
as many teeth as one has set with the 
corresponding peg S. Accordingly, 
in the above illustration, when we start at the zero position, and 
turn the handle once, the units wheel must jump to 2, the ten's 
wheel to 1, so that 12 appears. A second turn of the handle moves 
the units wheel another 2 and the tens wheel another 1, so that 24 ap- 
pears, and similarly, we get, after 3 or 4 times, 3 12 = 36 or 4 12 = 48, 


Fig. 3 a. 


Driving wheel 


Arithmetic: Calculating with Natural Numbers. 

But now turn the handle a fifth time: Again, according to the 
account above, the units wheel should jump again by two units, in other 
words back to 0, the tens wheel by one, or to 5, and we should have 
the false result 5 12 = 50. In the actual turning, however, the slide 
shows 50, to be sure, until just before the completion of the turn; but 
at the last instant the 5 changes into 6, so that the correct result appears. 
Something has come into action now that we have not yet described, 
and which is really the most remarkable point of such machines: the 
so called carrying the tens. Its principle is as follows: when one of the 
number bearing cogwheels under the slide (e. g., the units wheel) goes 
through zero, it presses an otherwise inoperative tooth of the neighboring 
driver (for the tens) into position, so that it engages the corresponding 
cogwheel (the tens wheel) and pushes this forward one place farther than 
it would have gone otherwise. You can understand the details of this 
construction only by examining the apparatus itself. There is the less 
need for my -going into particulars here because it is just the method 
of carrying the tens that is worked out in the greatest variety of ways 
in the different makes of machines, but I recommend a careful examina- 
tion of our machine as an example of a most ingenious model. Our 
collection contains separately the most important parts of the Brunsviga 
which are for the .most part invisible in the assembled machine so 
that you can, by examining them, get a complete picture of its ar- 

We can best characterize the operation of the machine, so far as 
we have made its acquaintance, by the words adding machine, because, 
with every turn of the handle, it adds, once, to the number on the slide at 
the right, the number which has been set on the drum. 

Finally, I shall describe in general that arrangement of the machine 
which permits convenient operation with multipliers of more than one 

digit. If we wish to calculate, say, 
15 12 we should have to turn the 
handle fifteen times, according to 
the plan already outlined; moreover, 
if one wished to have the multiplier 
indicated by the counter at the left 
of the slide, it would be necessary 
to have, there also, a device for 
carrying the tens. Both of these 

difficulties are avoided by the following arrangement 1 . We first perform 
the multiplication by five, so that 5 appears on the slide at the left 
and 60 at the right (see Fig. 5). Now we push the slide one place to the 


i ! 














Fig. 5. 

1 In the newer models the cogwheel device for "carrying over" is likewise 
very complete. . 

Practice in Calculating with Integers. 21 

right, so that, as shown in Fig. 5, its units cogwheel is cut out, its 
tens cogwheel is moved under the units slit of the drum, its hundreds 
cogwheel under, the tens slit, etc., while, at the left, this shift brings 
it about that the tens cogwheel, instead of the units, is connected 
with the driver which the handle carries. If we now turn the handle 
once, 1 appears at the left, in ten's place, so that we read 15; at the 

right, however, we do not get the addition | , .^ but | ^ or, in 

other words, 60 + 120, since the 2 is ' 'carried over" to the tens wheel, 
the 1 to the hundreds wheel. Thus we get correctly 15 12 = 180.. 
It is, as you see, the exact mechanical translation of the customary process 
of written multiplication, in which one writes down under one another, 
the products of the multiplicand by the successive digits of the 
multiplier, each product moved to the left one place farther than the 
preceding, and then adds. In just the same way one proceeds quite 
generally when the multiplier has three or more digits, that is, after the 
usual multiplication by the ones, one moves the slide 1,2, ... places to 
the right and turns the handle in each place as many times as the digit 
in the tens, hundreds, . . . place of the multiplier indicates. 

Direct examination of the machine will disclose how one can perform 
other calculations with it; the remark here will suffice that subtraction 
and division are effected by turning the handle in the direction opposite 
to that employed in addition. 

Permit me to summarize by remarking that the theoretical principle 
of the machine is quite elementary and represents merely a technical 
realization of the rules which one always uses in numerical calculation. 
That the machine really functions reliably, that all the parts engage 
one another with unfailing certainty, so that there is no jamming, that 
the wheels do not turn farther than is necessary, is, of course, the 
remarkable accomplishment of the man who made the design, and the 
mechanician who carried it out. 

Let us consider for a moment the general significance of the fact that 
there really are such calculating machines, which relieve the mathematician 
of the purely mechanical work of numerical calculation, and which do 
this work faster, and, to a higher degree free from error, than he himself 
could do it, since the errors of human carelessness do not creep into 
the machine. In the existence of such a machine we see an outright 
confirmation that the rules of operation alone, and not the meaning of 
the numbers themselves, are of importance in calculating; for it is only 
these that the machine can follow; it is constructed to do just that; 
it could not possibly have an intuitive appreciation of the meaning of 
the numbers. We shall not, then, wish to consider it as accidental that 
such a man as Leibniz, who as both an abstract thinker of first rank 
and a man of the highest practical gifts, was, at the same tine, both 

22 Arithmetic: The First Extension of the Notion of Number. 

the father of purely formal mathematics and the inventor of a calcu- 
lating machine. His machine is, to this day, one of the most prized 
possessions of the Kastner Museum in Hannover. Although it is not 
historically authenticated, still I like to assume that when Leibniz 
invented the calculating machine, he not only followed a useful purpose, 
but that he also wished to exhibit, clearly, the purely formal character 
of mathematical calculation. 

With the construction of the calculating machine Leibniz certainly 
did not wish to minimize the value of mathematical thinking, and yet it 
is just such conclusions which are now sometimes drawn from the 
existence of the calculating machine. If the activity of a science can be 
supplied by a machine, that science cannot amount to much, so it is 
said; and hence it deserves a subordinate place. The answer to such 
arguments, however, is that the mathematician, even when he is himself 
operating with numbers and formulas, is by no means an inferior counter- 
part of the errorless machine, ''thoughtless thinker" of Thomae; but 
rather, he sets for himself his problems with definite, interesting, and 
valuable ends in view, and carries them to solution in appropriate and 
original manner. He turns over to the machine only certain operations 
which recur frequently in the same way, and it is precisely the mathe- 
maticianone must not forget this who invented the machine for his 
own relief, and who, for his own intelligent ends, designates the tasks 
which it shall perform. 

Let me close this chapter with the wish that the calculating machine, 
in view of its great importance, may become known in wider circles 
than is now the case. Above all, every teacher of mathematics should 
become familiar with it, and it ought to be possible to have it demon- 
strated in secondary instruction. 

II. The First Extension of the Notion of Number 

With the last section we leave operations with integers, and shall 
treat, in a new chapter, the extension of the number concept. In the 
schools it is customary, in this field, to take in order the following steps : 

1. Introduction of fractions and operations with fractions. 

2. Treatment of negative numbers, in connection with the beginnings 
of operations with letters. 

3. More or less complete presentation of the notion of irrational numbers 
by examples that arise upon different occasions, which leads, then, gra- 
dually, to the notion of the continuum of real numbers. 

It is a matter of indifference in which order we take up the first 
two points. Let us discuss negative numbers before fractions. 

Negative Numbers. 2} 

1. Negative Numbers 

Let us first note, as to terminology, that in the schools, one speaks 
of positive and negative numbers, inclusively, as relative numbers in 
distinction from the absolute (positive) numbers, whereas, in universities 
this language is not common. Moreover, in the schools one speaks of 
"algebraic numbers" 1 along with relative numbers, an expression which 
we in universities employ, as you know, in quite another sense. 

Now, as to the origin and introduction of negative numbers, I can 
be brief in my reference to source material; these things are already 
familiar to you, or you can at least easily make them so with the help 
the references I shall give. You will find a complete treatment, for 
example, in Weber- Wellstein ; also, in very readable form, in H. Burk- 
hardt's Algebraischer Analysis 2 . This book, moreover, you might well 
purchase, as it is of moderate size. 

The creation of negative numbers is motivated, as you know, by 
the demand that the operation of subtraction shall be possible in all cases. 
If a < b then a b is meaningless in the domain of natural integers; 
a number c = b a does exist, however, and we write 

a b = c 

which we call a negative number. This definition at once justifies the 
representation of all integers by means of the scale of equidistant points 

I 1 1 1 1 1 1 1 1 

A +1 +2 +3 +4 

on a straight line the "axis of abscissas" which extends in both directions 
from an pjip 1 "". Gn^ A a y rtr msider this picture as a common possession 
of all educated persons today, and one can, perhaps, assume that it 
owes its general dissemination, chit^y, to the thermometer scale. The 
commercial balance, with its reckonii _ 'n debits and credits, affords 
likewise a graphic and familiar picture of negative numbers. 

Let us, however, realize at once and emphatically how extr" ^Jiu " v 
difficult in principle is the step, which is taken in school when negative 
numbers are introduced. Where the pupil before was accustomed to 
represent visually by concrete numbers of things the numbers, and, 
later, the letters, with which he operated, as well as the results which 
he obtained by his operations, he finds it now quite different. He has 
to do with something new, the "negative numbers", which have, imme- 
diately, nothing in common with his picture of numbers of things, but 
he must operate with them as though they had, although the operations 

1 See, e. g. Mehler, Hauptsatze der Elementarmathematik, Nineteenth edition, 
p. 77, Berlin, 1895- 

2 Leipzig 1903. [Third edition, revised by G. Faber, 1920.] See also Fine, H., 
The Number-System of Algebra treated Theoretically and Historically, Heath. 

24 Arithmetic: The First Extension of the Notion of Number. 

have graphically a meaning much less clear than the old ones. Here, 
for the first time, we meet the transition from concrete to formal mathe- 
matics. The complete mastery of this transition requires a high order 
of ability in abstraction. 

We shall now inquire in detail what happens to the operations of 
calculation when negative numbers are introduced. The first thing to 
notice is that addition and subtraction coalesce, substantially: The 
addition of a positive number is the subtraction of the equal and opposite 
negative number, In this connection, Max Simon makes the amusing 
remark that, whereas negative numbers were created to make the 
operation of subtraction possible without any exception, subtraction as 
an independent operation ceased to exist by virtue of that creation. 
For this new operation of addition (including subtraction) in the domain 
of positive and negative numbers the five formal laws stated before 
hold without change. These are, in brief (see p. 9 et seq.) : 

1. Always possible. 

2. Unique. 

3. Associative law. 

4. Commutative law. " 

5. Monotonic law. 

Notice, in connection with 5, that a < b means, now, that a lies to 
the left of b in the geometric representation, so that we have, for 
example 2 < 1 , 3 < 2. 

The chief point in the multiplication of positive and negative numbers 
is the rule of signs, that a - ( c) = ( c) a = (a c), and ( c) ( c') 
= + (c c'). Especially the latter rule: ' 'Minus times minus gives plus" 
is often a dangerous stumbling block. \K* shall return preaymrjy to the 
inner significance of these rules; jn^ J *iow we shall combine them into 
a statement defining multiplv^u&n f a series of positive and negative 
numbers: The absolute val* e of a product is equal to the product of the 
afarfute values of the.*-t rs >' ^ s sign is positive or negative according as 
an even or an oaa number of factors is negative. With this convention, 
multiplication in the domain of positive and negative numbers has again 
the following properties: 

1. Always possible. 

2. Unique. 

3. Associative. 

4. Commutative. 

5. Distributive with respect to addition. 

There is a change only in the monotonic law; in its place one has 
the following law: 

6. If a > b then a - c ^ b c according as c 5^ 0. 

Let us inquire, now, whether these laws, considered again purely 
formally, are consistent. We must admit at once, however, that a purely 

Negative Numbers. 25 

logical proof of consistency is as yet much less possible here than it is 
in the case of integers. Only a reduction is possible, in the sense that 
the present laws are consistent if the laws for integers are consistent. 
But until this has been completed by a logical consistency proof for 
integers, one will have to hold that the consistency of our laws is based 
solely on the fact that there are intuitive things, with intuitive relations, 
which obey these laws. We noted above, as such, the series of integral 
points on the axis of abscissas and we need only indicate what the rules 
of operation signify there: The addition %' = x + #, where a is fixed, 
assigns to each point x a second point x', so that the infinite straight 
line is simply displaced along itself by an amount a , to the right or to 
the left, according as a is positive or negative. In an analogous manner, 
the multiplication x' = a % represents a similarity transformation of 
the line into itself, a pure stretching for a > , a stretching together 
with a reflexion in the origin for a < . 

Permit me now to explain how, historically, all these things arose. 
One must not think that the negative numbers are the invention of 
some clever man who menufactured them, together with their con- 
sistency perhaps, out of the geometric representation. Rather, during 
a long period of development, the use of negative numbers forced itself, 
so to speak, upon mathematicians. Only in the nineteenth century, 
after men had been operating with them for centuries, was the con- 
sideration of their consistency taken up. 

Let me preface the history of negative numbers with the remark 
that the ancient Greeks certainly had no negative numbers, so that 
one cannot yield them the first place, in this case, as so many people 
are otherwise prone to do. One must attribute this invention to the 
Hindus, who also created our system of digits and in particular our zero. 
In Europe, negative numbers came gradually into use at the time of 
the Renaissance, just as the transition to operating with letters had 
been completed. I must not omit to mention here that this completion 
of operations with letters is said to have been accomplished by Vieta 
in his book In Artem Analyticam Isagoge 1 . 

From the present point of view, we have the so called parenthesis 
rules for operations with positive numbers, which are, of course, con- 
tained in our fundamental formulas, provided one includes the correpond- 
ing laws for subtraction. But I should like to take them up somewhat 
in detail, by means of two examples, in order, above all, to show the 
possibility of extremely simple intuitive proofs for them, proofs which 
need consist only of the representation and of the word "Look"!, as 
was the custom with the ancient Hindus. 

\. Given a > b and c > a, where a t b,c are positive. Then a b 
is a positive number and is smaller than c , that is, c (a b) must 

1 Tours 1591. 

26 Arithmetic: The First Extension of the Notion of Number. 

exist as a positive number. Let us represent the numbers on the axis 
of abscissas and note that the segment between the points b and a has 
the length a b. A glance at the representation shows that, if we 
take away from c the segment a 6, the result is the same as though 
we first took away the entire segment a and then restored the part &, i. e., 

(1) c (a b) = c a + b . 

2. Given a >6 and c > d; then a b and c d are positive integers. 
We wish to examine the product (a b) (c d) ; for that purpose 
1 1 1 1 

b _ a c 

a^b ' 

draw the diagonally hatched rectangle (Fig. 6) with sides a b and 
c d whose area is the number sought, (a b) (c d) , and which 
is part of the rectangle with sides a and c . In order to obtain the former 
rectangle from the latter, we take away first the horizontally hatched 

rectangle a d, then the vertically 
J L " hatched one b-c; in doing this we 

have removed twice the double-hatched 

rectangle b d , and we must put it back. 

But these operations express precisely 

the known formula 

(2) (a b)(c d) = ac ad bc + bd. 

As the most important psycholog- 
ical moment to which the introduction 
of negative numbers, upon this basis of 

operations with letters, gave rise, that general peculiarity of human 
nature shows itself, by virtue of which we are involuntarily inclined to 
employ rules under circumstances more general than are warranted by the 
special cases under which the rules were derived and have validity. This was 
first claimed as a guiding principle in arithmetic by Hermann Hankel, in 
his Theorie der komplexen Zahlsysteme* 1 , under the name "Prinzip von 
der Permanenz der formalen Gesetze" **. I can recommend to your notice 
this most interesting book. For the particular case before us, of transition 
to negative numbers, the above principle would declare that one desired 
to forget, in formulas like (1) and (2) the expressed assumptions as to 
the relative magnitude of a and b and to employ them in other cases. 
If one applies (2), for example, to a = c = 0, for which the formulas 
were not proved at all, one obtains ( b) ( d) = + bd, i. e., the sign 
rule for multiplication of negative numbers. In this manner we may 
derive, in fact almost unconsciously, all the rules, which we must now 

* Theory of Complex Number Systems. 
1 Leipzig 1867. 
** Principle of the permance of formal laws. 

Negative Numbers. 27 

designate, following the same line of thought, as almost necessary as- 
sumptions, necessary insofar as one would have validity of the old rules 
for the new concepts. To be sure, the old mathematicians were not happy 
with this abstraction, and their uneasy consciences found expression in 
names like invented numbers, false numbers, etc., which they gave to 
the negative numbers on occasion. But , in spite of all scruples, the 
negative numbers found more and more general recognition in the 
sixteenth and seventeenth centuries, because they justified themselves 
by their usefulness. To this end, the development of analytic geometry 
without doubt contributed materially. Nevertheless the doubts per- 
sisted, and were bound to persist, so long as one continued to seek for 
a representation in the concept of a number of things, and had not 
recognized the leading role of formal laws when new concepts are set 
up. In connection with this stood the continually recurring attempts 
to prove the rule of signs. The simple explanation, which was brought 
out in the nineteenth century, is that it is idle to talk of the logical 
necessity of the theorem, in other words, the rule of signs is not 
susceptible of proof] one can only be concerned with recognizing the 
logical permissibility of the rule, and, at the same time, that it is 
arbitrary, and regulated by considerations of expedience, such as the 
principle of permanence. 

In this connection one cannot repress that oft recurring thought 
that things sometimes seem to be more sensible than human beings. 
Think of it: one of the greatest advances in mathematics, the intro- 
duction of negative numbers and of operations with them, was not 
created by the conscious logical reflection of an individual. On the 
contrary, its slow organic growth developed as a result of intensive 
occupation with things, so that it almost seems as though men had 
learned from the letters. The rational reflection that one devised here 
something correct, compatible with strict logic, came at a much later 
time. And, after all, the function of pure logic, when it comes to setting 
up new concepts, is only to regulate and never to act as the sole guiding 
principle', for there will always be, of course, many other conceptual 
systems which satisfy the single demand of logic, namely, freedom from 

If you desire still other literature concerning questions about the 
history of negative numbers, let me recommend Tropfkes Geschichte der 
Elementarmathematik 1 * , as an excellent collection of material containing, 
in lucid presentation, a great many details about the development of 
elementary notions, views, and names. 

1 Two volumes, Leipzig 1902/03- [Second edition revised and much enlarged, to 
appear in seven volumes, of which six had appeared by 1924.] See also Cajori, F. ( 
History of Mathematics, Macmillan. 

* History of Elementary Mathematics. 

2g Arithmetic: The First Extension of the Notion of Number. 

If we now look critically at the way in which negative numbers 
are presented in the schools, we find frequently the error of trying to 
prove the logical necessity of the rule of signs, corresponding to the 
above noted efforts of the older mathematicians. One is to derive 
(6) ( d) = +bd heuristically, from the formula (a b) (c d) and 
to think that one has a proof, completely ignoring the fact that the 
validity of this formula depends on the inequalities a > 6, c > d 1 . Thus 
the proof is fraudulent, and the psychological consideration which would 
lead us to the rule by way of the principle of permanence is lost in favor 
of quasi-logical considerations. Of course the pupil, to whom it is thus 
presented for the first time, cannot possibly comprehend it, but in the end 
he must nevertheless believe it; and if, as it often happens, the repeti- 
tion in a higher class does not supply the corrective, the conviction may 
become lodged with some students that the whole thing is mysterious, 

In opposition to this practice, I should like to urge you, in general, 
never to attempt to make impossible proofs appear valid. One should 
convince the pupil by simple examples, or, if possible, let him find out 
for himself that, in view of the actual situation, precisely these con- 
ventions, suggested by the principle of permanence, are appropriate in that 
they yield a uniformly convenient algorithm, whereas every other convention 
would always compel the consideration of numerous special cases. To be 
sure, one must not be precipitate, but must allow the pupil time for 
the revolution in his thinking which this knowledge will provoke. And 
while it is easy to understand that other conventions are not advanta- 
geous, one must emphasize to the pupil how really wonderful the fact 
is that a general useful convention really exists ; it should become clear 
to him that this is by no means self-evident. 

With this I close my discussion of the theory of negative numbers 
and invite you now to give similar consideration to the second extension 
of the notion of number. 

2. Fractions. 

Let us begin with the treatment of fractions in the schools. There 
the fraction a/b has a thoroughly concrete meaning from the start. In 
contrast to the graphic picture of the integer, there has been only a 
change of base: We have passed from the number of things to their 
measure, from the consideration of countable things to measurable things. 
The system of coins, or of weights, affords, with some restriction, and 
the system of lengths affords completely, an example of measurable mani- 
folds. These are the examples with which the idea of the fraction is 

1 See, for example, . Heis, Sammlung von Beispielen und Aufgaben aus der 
Arithmetik und Algebra. Edition 1904, p. 46, 106108. 

Fractions. 29 

given to every pupil. No one has great difficulty in grasping the meaning 
of x / 3 meter oder x / 2 pound. The relations =,>,<, between fractions 
can be immediately developed by means of the same concrete intuition, 
and likewise the operations of addition and subtraction, as well as the 
multiplication of a fraction by an integer. After this, general multiplication 
can easily be made ofeprehensible : To multiply a number by a/b means 
to multiply it by a ana then to divide by 6; in other words: the product is 
derived from the multiplicand just as a/b is derived from \ . Division by 
a fraction is then presented as the operation inverse to multiplication: 
a divided by 2/3 is the number which multiplied by 2/3 gives a. These 
notions of operations with fractions combine with that of negative 
numbers so that one finally has the totality of all rational numbers. 
I cannot enter into the details of this building-up process, which, in the 
school, takes, of course, a long time. Let us rather compare it at once 
with the perfected presentation of modern mathematics, using for 
this purpose the above mentioned books of Weber- Wellstein and 
Burkhardt 1 . 

Weber-Wellstein emphasizes primarily the formal point of view which, 
from the multiplicity of possible interpretations, selects what is of 
necessity common to all. According to this view, the fraction a/b is 
a symbol, a "number-pair" with which one can operate according to 
certain rules. These rules, which in our discussion above arose naturally 
from the meaning of fraction, have here the character of arbitrary con- 
ventions. For example, that which, to the pupil, is an obvious theorem 
concerning the multiplication or division of both terms of a fraction 
by the same number, appears here as a definition of equality: two 
fractions a/b, c/d are called equal when ad = be. Similarly, greater than 

and smaller than are defined, and one agrees that the fraction ( j--j ) 

shall be called the sum of the two fractions a/b, c/d, etc. It is thus proved 
that the operations, so defined in the new domain of numbers, possess 
formally exactly the properties of addition and multiplication for in- 
tegers, i. e., they satisfy the eleven fundamental laws which have been 
repeatedly enumerated. 

Burkhardt does not proceed quite so formally as does Weber-Well- 
stein, whose presentation we have sketched in its essentials. He looks 
upon the fraction a/b as a sequence of two operations in the domain of 
integers: a multiplication by a and a division by b, in which the object 
upon which these operations are performed is an arbitrarily chosen 
integer. If one undertakes two such "pairs bf operations" a/b, c/d, this 
is said to correspond to multiplication of the fractions, and one sees easily 
that the operation so resulting is none other than multiplication by a c 
and division by b d, so that the rule for the multiplication of fractions, 

1 In what follows, the first editions of these books have been used. 

Arithmetic: The First Extension of the Notion of Number. 

y) " \~d) ~ \b^~d)' ^toincd ou ^ f ^ e c ^ ear meaning of the fractions, 
but not determined merely as an arbitrary convention. One can, of 
course, treat division in the same way. Addition and subtraction, on 
the other hand, do not admit of such a simple explanation with this 

representation ; thus the formula -j- + -T- = , ,^ remains, with Burk- 

hardt also, only a convention for which he adtfuces only reasons of 

Let us now compare the older presentation in the schools, with the 
modern conception just sketched. According to the latter, in the one 
book as well as in the other, we are left really completely in the field of 
integers, in spite of the extension of the notion of number. It is merely 
assumed that the totality of whole numbers is intuitively grasped, or 
that the rules of operation with them are known; the things newly 
defined as number-pairs, or as operations with whole numbers, fit 
completely into this frame. The school treatment, on the other hand, 
is based entirely on the newly acquired conception of measurable quan- 
tities, which supplies an immediate intuitive picture of fractions. We 
can best grasp this difference if we imagine a being who has the notion 
of whole numbers, but no conception of measurable quantities. For him 
the school presentation would be wholly unintelligible, whereas he could 
well comprehend the discussions of either Weber- Wellstein or Burkhardt. 

Which of the two methods is the better? What does each accomplish? 
The answer to this will be like the one we gave recently when we put 
the analogous question concerning the different conceptions of integers. 
The modern presentation is surely purer, but it is also less rich. For, 
of that which the traditional curriculum supplies as a unit, it gives 
really only one part : the abstract and logically complete introduction 
of certain arithmetic concepts, called "fractions" , and of operations with 
them. But it leaves unexplained an entirely independent and no less 
important question: Can one really apply the theoretical doctrine so 
derived, to the concrete measurable quantities about us? Again one 
could call this a problem of "applied mathematics", which admits an 
entirely independent treatment. To be sure, it is questionable whether 
such a separation would be desirable pedagogically. In Weber- Wellstein, 
moreover, this splitting of the problem into two parts finds characteristic 
expression. After the abstract introduction of operations with fractions, 
of which alone we have thus far taken account, they devote a special 
(the fifth) division called "ratios" to the question of applying rational 
numbers to the external world. The presentation is, to be sure, rather 
abstract than intuitive. 

I shall now close this discussion of fractions with a general remark 
concerning the totality of rational numbers, where, for the sake of 
clearness, I shall make use of the representation upon a straight line. 

Irrational Numbers. if 

Think of all points with rational abscissas marked upon this line; we 
designate them briefly as rational points. We say, then, that the totality 
of these rational points on the axis of abscissas is "dense", meaning 
that in every interval, however small, there are still infinitely many 

. . . I . . . . . I I , . , , , I , , 

rational points. If we wish to avoid putting anything new into the 
notion of rational numbers, we might say, more abstractly, that between 
any two rational points there is always another rational point. It follows 
that one can separate from the totality of rational points, finite parts 
which contain neither a smallest nor a largest element. The totality 
of all rational points between and 1 , these points excluded, is an 
example. For, given any number between and 1 , there would still 
be a number between it and 0, i. e., a smaller, and a number between 
it and 1 , i. e., a larger. In their systematic development, these concepts 
belong to the theory of point sets of Cantor. In fact, we shall make use 
later of the totality of rational numbers, together with the property 
just mentioned, as an important example of a point set. 

I shall pass now to the third extension of the number system: the 
irrational numbers. 

3. Irrational Numbers. 

Let us not spend any time in discussing how this field is usually 
treated in the schools, for there one does not get much beyond a few 
examples. Let us rather proceed at once to the historical development. 
Historically, the origin of the concept of irrational 
numbers lies certainly in geometric intuition and in 
the requirements of geometry. If we consider, as 
we did just now, that the set of rational points is 
dense on the axis of abscissas, then there are still 

other points on it. Pythagoras is said to have shown p . 

this in a manner somewhat as follows. Given a right 

triangle with each leg of length 1 , then the hypotenuse is of length 

}/2, and this is certainly not a rational number; for if one puts y 2 = -r 

where a and b are integers, prime to each other, one is led easily by the 
laws of divisibility of integers to a contradiction. // we now lay off 
geometrically on the axis of abscissas, beginning at zero, the segment thus 
constructed, we obtain a non-rational point which is not one of the original 
set that is dense on the axis. Furthermore, the Pythagoreans certainly 
were aware that, in most cases, the hypotenuse, 1/m 2 + n 2 , of a right 
triangle with legs m and n t is irrational. The discovery of this extra- 
ordinarily essential fact was indeed worth the sacrifice of one hundred 

12 Arithmetic: The First Extension of the Notion of Number. 

oxen with which Pythagoras is said to have celebrated it. We know, also 
that the Pythagorean School was fond of searching out those special pairs 
of values for m and n for which the right triangle has three commensurable 
sides, whose lengths, in an appropriately selected unit of measure, can 
be expressed in integers (so called Pythagorean numbers). The simplest 
example of one of these number-triples is '3, .4&5 

Later Greek mathematicians studied, in addition to these simplest 
irrationalities, others that were more complicated; thus one finds in 

Euclid types such as }//0 + /&, and the like. We may say, however, 
in general, that they confined themselves essentially to such irrationali- 
ties as one obtains by repeated extraction of square root, and which 
can therefore be 1 constructed geometrically with ruler and compasses. 
The general idea of irrational number was not yet known to them. 

I must, modify this remark somewhat, however, in order to avoid 
misunderstanding. The more precise statement is that the Greeks 
possessed no method for producing or defining, arithmetically, the 
general irrational number in terms of rational numbers. This is a result 
of modern development and will soon engage our attention. Nevertheless, 
from another point of view they were familiar with the notion of the 
general real number which was not necessarily rational; but the concept 
had an entirely different appearance to them because they did not use 
letters for general numbers. In fact they studied, and Euclid developed 
very systematically, ratios of two arbitrary segments. They operated with 
such ratios precisely as we do today with arbitrary real numbers. In- 
deed we find in Euclid definitions which suggest strongly the modern 
theory of irrational numbers. Moreover the name used is different from 
that of the natural number; the latter is called ciQiftfios , whereas the 
line ratio, the arbitrary real number, is called Myos. 

I should like to add a remark concerning the word "irrational". It 
is without doubt the translation into Latin of the Greek "fiyloj'og". 
The Greek word, however, meant presumably "inexpressible" and im- 
plied that the new numbers, or line ratios, could not, like the rational 
numbers, be expressed by the ratio of two whole numbers 1 . The 
misunderstanding put upon the Latin "ratio", that it could convey 
only the meaning "reason", gave to "irrational" the meaning "unreaso- 
nable", which seems still to cling to the term irrational number. 

The general idea of the irrational number appeared first at the end 
of the sixteenth qentury as a consequence of the introduction of decimal 
fractions, the use of which became established at that time in connection 
with the appearance of logarithmic tables. If we transform a rational 
number into a decimal, we may obtain infinite decimals*, as well as finite 

1 See Tropfke, second edition, Vol. 2, p. 71. 

2 For complete treatment of this subject see, p. 40 et seq. 

Irrational Numbers. 


decimals, but they will always be periodic. The simplest example is 
J = 0.333 . . . , i.e. , a decimal whose period of one digit begins imme- 
diately after the decimal point. Now there is nothing to prevent our 
thinking of an aperiodic decimal whose digits proceed according to any 
definite law whatever, and anyone would instinctively consider it as 
a definite, and hence|& non-rational, number. By this means the general 
notion of irrational number is established. It arose to a certain extent 
automatically, by the consideration of decimal fractions. Thus, histori- 
cally, the same thing happened with irrational numbers that, as we have 
seen, happened with negative numbers. Calculation forced the intro- 
duction of the new concepts, and without being concerned much as to 
their nature or their motivation, one operated with them, the more 
particularly since they often proved to be extremely useful. 

It was not until the sixth decade of the nineteenth century that the 
need was felt for a more precise arithmetic formulation of the foun- 
dations of irrational numbers. This occurred in the lectures which 
Weierstrass delivered at about that date. In 1872, a general foundation 
was laid simultaneously by G. Cantor of Halle, the founder of the theory 
of point sets, and independently by R. Dedekind of Braunschweig. I 
will explain Dedekind' s point of view in a few words. Let us assume 
a knowledge of. the totality of rational numbers, but let us exclude 
all space perception, which would force upon us forthwith the notion of 
the continuity of the number series. With this understanding, in order 
to attain to a purely arithmetic definition of the irrational number, 
Dedekind sets up the notion of a "cut" in the domain of rational numbers. 
If r is any rational number, it separates the totality of rational numbers 
into two parts A and B such thai ev$ry number in A is smaller than any 
number in B and every rational number belongs to one of these two classes. 
A is the totality of all rational numbers which are smaller than r , B those 
that are larger, whereby r itself may be thought of indifferently as be- 
longing to the one or to the other. Besides these "proper cuts" there are 
also "improper cuts", these being separations of all rational numbers 
into two classes having the same properties except that they are not 
brought about by a rational number, i. e., separations such that there 
is neither a smallest rational number in B nor a largest in A . An example 

of such an improper cut is supplied by, say, "^2 = 1.414 ... In fact, 
every infinite decimal fraction defines a cut, provided one assigns to B 
every rational number which is larger than every approximation to the 
infinite decimal, and to A every other rational number ; each number 
in A would thus be equalled or exceeded by at least one approximation 
(and hence by infinitely many). One can easily show that this cut is 
proper if the decimal is periodic, improper if it is not periodic. 

With these considerations as his basis, Dedekind sets up his definition, 
which, from a purely logical standpoint, must be looked upon as an 

Klein, Elementary Mathematics. 3 

^4 Arithmetic: The First Extension of the Notion of Number. 

arbitrary convention : A cut in the domain of rational numbers is called 
a rational number or an irrational number according as the cut is proper 
or improper. A definition of equality follows from this at once: Two 
numbers are said to be equal if they yield the same cut in the domain of 
rational numbers. From this definition we can immediately prove for 
example, that, J / 3 is equal to the infinite decimal 0.^333 .... If we accept 
this standpoint, we must demand a proof, i. e., a process of reasoning 
depending upon the definition given, although this would appear quite 
unnecessary to one approaching the subject naively. Moreover, such 
a proof is immediate, if one reflects that every rational number smaller 
than x / 3 will be exceeded ultimately by the decimal approximations, 
whereas these are smaller than every rational number which exceeds J. 
The corresponding definition in the lectures of Weierstrass appears in 
the following form: Two numbers are called equal if they differ* by less 
than any preassigned constant, however small. The connection with the 
preceding explanation is clear. The last definition becomes striking if 
one reflects why 0.999 is equal to 1 ; the difference is certainly 
smaller than 0.1, smaller than 0.01, etc., that is, it is exactly zero, 
according to the definition. 

If we enquire how it happens that we can admit the irrational 
numbers into the system of ordinary numbers and operate with them 
in just the same way, the answer is to be found in the validity of the 
monotonic law for the four fundamental operations. The principle is as 
follows: // we wish to perform upon irrational numbers the operation of 
addition , multiplication, etc., we can enclose them between ever narrowing 
rational limits and perform upon these limits the desired operations ; then, 
because of the validity of the monotonic law, the result will also be enclosed 
between ever narrowing limits. 

It is hardly necessary for me to explain these things in greater 
detail, since very readable presentations of them are easily available in 
many books, especially in Weber-Wellstein and in Burkhardt. I hope 
that you will read more fully than I could tell you here in these books, 
about the definition of irrational numbers. 

I should prefer, rather, to talk about something which you will 
hardly find in the books, namely, how, after establishing this arithmetic 
theory, we can pass to the applications in other fields. This applies in 
particular, to analytic geometry, which to the naive perception appears 
to be (and psychologically really is) the source of irrational numbers. 
If we think of the axis of abscissa, with the origin and also the rational 
points marked on it, as above, then these applications depend upon 
the following fundamental principle: Corresponding to every rational or 
irrational number there is a point which has this number as abscissa and, 
conversely, corresponding to every point on the line there is a rational or 
an irrational number, viz., its abscissa. Such a fundamental principle, 

Irrational Numbers. 


which stands at the head of a branch of knowledge, and from. which 
all that follows is logically deduced, while it itself cannot be logically 
proved, may properly be called an axiom. Such an axiom will appear 
intuitively obvious or will be accepted as a more or less arbitrary con- 
vention, by each person according to his gifts. This axiom concerning 
the one-to-one correspondence between real numbers on one hand, and 
the points of a straight line on the other, is usually called the Cantor 
axiom because G. Cantor was the first to formulate it specifically (in 
the Mathematische Annalen, vol. 5, 1872). 

This is the proper place to say a word about the nature of space 
perception. It is variously ascribed to two different sources of knowledge. 
One the sensibly immediate, the empirical intuition of space, which we 
can control by means of measurement. The other is quite different, 
and consists in a subjective idealizing intuition, one might say, perhaps, 
our inherent idea of space, which goes beyond the inexactness of sense 
observation. I pointed out to you an analogoiis difference when we were 
discussing the notion of number. We may characterize it best as follows: 
It is immediately clear to us what a small number means, like 2 or 5, 
or even 7, whereas we do not have such immediate intuition of a larger 
number, say 2503- Immediate intuition is replaced here by the sub- 
jective intuition of an ordered number series, which we derive from 
the first numbers by mathematical induction. There is a similar situation 
regarding space perception. Thus, if we think of the distance between 
two points, we can estimate or measure it only to a limited degree of 
exactness, because our eyes cannot recognize as different two line-segments 
whose difference in length lies below a certain limit. This is the concept 
of the threshold of perception which plays such an important role in 
psychology. This phenomenon still persists, in its essentials, when we 
aid the eye with instruments of the highest precision; for there are 
physical properties which prohibit our exceeding a certain degree of 
exactness. For instance, optics teaches that the wave-length of light, 
which varies with the color, is of the order of smallness of 1 / 10 oo mm - 
(= 1 micron); it shows also that objects whose dimensions are of this 
order of smallness cannot be seen distinctly with the best microscopes 
because diffraction enters then and hence no optical image can give 
exact reproductions of the details. The result of this is the impossibility, by 
direct optical means, of getting measures of length that are finer than to 
within one micron, so that, when measured lengths are given in millimeters, 
only the first three decimals can have an assured meaning. In the same 
way, in all physical observations and measurements, one meets such 
threshold values which cannot be passed, which determine the extreme 
limits of possible exactness of lengths which have been measured and 
expressed in millimeters. Statements beyond this limit have no meaning, 
and are an evidence of ignorance or of attempted deception. One often 

3 () Arithmetic: The First Extension of the Notion of Number. 

finds such excessively exact numbers in the advertisements of medicinal 
springs, where the percentage of salt, which really varies with the 
time, is given to a number of decimal places which could not possibly 
be determind by weighing. 

In contrast with this property of empirical space perception which 
is restricted by limitations on exactness, abstract, or ideal space perception 
demands unlimited exactness, by virtue of which, in view of Cantor's axiom, 
it corresponds exactly to the arithmetic definition of the number concept. 

In harmony with this division of our perception, it is natural to 
divide mathematics also into two parts, which have been called mathe- 
matics of approximation and the mathematics of precision. If we desire 
to explain this difference by an interpretation of the equation / (x) = 0, 
we may note that, in the mathematics of approximation, just as in our 
empirical space perception, one is not concerned that / (x) should be 
exactly zero, but .merely that its absolute value |/ (x)\ should remain 
below the attainable threshold of exactness . The symbol / (x) = is 
merely an abbreviation for the inequality | / (x) \ <C e , with which one 
is really concerned. It is only in the mathematics of precision that one 
insists that the equation / (x) = be exactly satisfied. Since mathe- 
matics of approximation alone plays a r61e in applications, one might 
say, somewhat crassly, that one needs only this branch of mathematics, 
whereas the mathematics of precision exists only for the intellectual 
pleasure of those who busy themselves with it, and to give valuable 
and indeed indispensable support for the development of mathematics 
of approximation. 

In order to return to our real subject, I add here the remark that 
the concept of irrational number belongs certainly only to mathematics of 
precision. For, the assertion that two points are separated by an ir- 
rational number of millimeters cannot possibly have a meaning, since, 
as we saw, when our rigid scales are measured in meters, all decimal 
places beyond the sixth are devoid of meaning. Thus in practice we can, 
without concern, replace irrational numbers by rational ones. This may 
seem, to be sure, to be contradicted by the fact that, in crystallography, 
one talks of the law of rational indices, or by the fact that in astronomy, 
one distinguishes different cases according as the periods of revolution 
of two planets have a rational or an irrational ratio. In reality, however, 
this form of expression only exhibits the many-sidedness of language ; 
for one is using here rational and irrational in a sense entirely different 
from that hitherto used, namely, in the sense of mathematics of approxi- 
mation. In this sense, one says that two magnitudes have a rational 
ratio when they are to each other as two small integers, say 3/7; whereas 
one would call the ratio 2021/7053 irrational. We cannot say how large 
numerator and denominator in this second case must be, in general, 
since that depends upon the problem in hand. I discussed all these 

Number Theory in the Schools. 17 

interesting relations in a course of lectures in the Summer Semester 
of i90i, wmcn was lithographed in 1902 and which will constitute the 
third volume of the present work (see the preface to the third edition, 
p. V) : Applications of Differential and Integral Calculus to Geometry, 
a Revision of Principles [Elaborated by C. H. Muller]. 

In conclusion let me say, in a few words, how I would have these 
matters handled in the schools. An exact theory of irrational numbers 
would hardly be adapted either to the interest or to the power of com- 
prehension of most of the pupils. The pupil will usually be content 
with results of limited exactness. He will look with astonished approval 
upon correctness to within 1 / 1000 mm and will not demand unlimited 
exactness. For the average pupil it will be sufficient if one makes the 
irrational number intelligible in general by means of examples, and 
this is what is usually done. To be sure, especially gifted individual 
pupils will demand a more complete explanation than this, and it will 
be a laudable exercise of pedagogical skill on the part of the teacher 
to give such students the desired supplementary explanation without 
sacrificing the interests of the majority. 

III. Concerning Special Properties of Integers 

We shall now begin a new chapter which will be devoted to the 
actual theory of integers, to the theory of numbers, or arithmetic in its 
narrower sense. I shall first recall in tabular form the individual ques- 
tions from this science which appear in the school curriculum. 

1 . The first problem of the theory of numbers is that of divisibility : 
Is one number divisible by another or not? 

2. Simple rules can be given which enable us easily to decide as to 
the divisibility of any given number by smaller numbers, such as 2, 3, 4, 
5, 9, 11, etc. 

3. There are infinitely many prime numbers, that is, numbers which 
have no integral divisors except one and themselves) : 2, 3, 4, 5, 9, 11, etc. 

4. We are in control of all of the properties of given integers if we 
know their decomposition into prime factors. 

5. In the transformation of rational fractions into decimal fractions 
the theory of numbers plays an important role ; it shows why the decimal 
fraction must be periodic and how large the period is. 

Although such questions may be considered in secondary schools, 
when the pupils are' between the ages of eleven and thirteen, the theory 
of numbers comes up only in isolated places during the later years, 
and, at most, the following points are considered. 

6. Continued fractions are taught occasionally, although not in all 

7. Sometimes instruction is given also in Diophantine equations, that 
is, equations with several unknowns which can take only integral values. 

38 Arithmetic: Concerning Special Properties of Integers. 

The Pythagorean numbers of which we spoke (see p. 32), furnish an 
example; here one has to do with triplets of integers which satisfy the 

a * + b* = c*. 

8. The problem of dividing the circle into equal parts is closely related 
to the theory of numbers, although the connection is hardly ever worked 
out in the schools. If we wish to divide the circle into n equal parts, 
using, of course, only ruler and compasses, it is easy to do it for n = 2, 3> 
4, 5,6. It cannot be done, however, if n = 7, hence we stop respect- 
fully when we come to this problem in the school. To be sure, it is not 
always stated definitely that -this construction is really impossible when 
n = 7, a fact whose explanation lies somewhat deep in number-theo- 
retic considerations. In order to forestall misunderstandings, which un- 
fortunately often arise, let me say, with emphasis, that one is concerned 
here again with a problem of mathematics of precision, which is devoid 
of meaning for the applications. In practice, even in cases where an 
"exact" construction is possible, it would not be used ordinarily; for, 
in the field of mathematics of approximation, the circle can be divided 
into any desired number of equal parts more suitably by simple skillful 
experiment ; and any prescribed, practically possible, degree of exactness 
can be attained. Every mechanician who makes instruments that carry 
divided circles proceeds in this way. 

9. The higher theory of numbers is touched by the school curriculum 
in one other place, namely, when n is calculated, during the study of the 
quadrature of the circle. We usually determine the first decimal places 
for &, by some method or other, and we mention incidentally, perhaps, 
the modern proof of the transcendence of n which sets at rest the old problem 
of the quadrature of the circle with ruler and compasses. At the end of 
this course I shall consider this proof in detail. For the present I shall 
give merely a prescise formulation of the fact, namely, that the number 
n does not satisfy any algebraic equation with integral coefficients: 

l + ... +kji+ \ -0. 

It is especially important that the coefficients be integers, and it is for 
this reason that the problem belongs to the theory of numbers. Of 
course here, again, one is concerned solely with a problem of the mathe- 
matics of precision, because it is only in this sense that the number- 
theoretic character of n has any significance. The mathematics of 
approximation is satisfied with the determination of the first few 
decimals, which permit us to effect the quadrature of the circle with 
any desired degree of exactness. 

I have sketched for you the place of the theory of numbers in 
the schools. Let us consider now its proper place in university instruction 
and in scientific investigation* In this connection I should like to divide 

Number Theory in the University. ^9 

research mathematicians, according to their attitude toward theory of 
numbers, into two classes, which I might call the enthusiastic class and 
the indifferent class. For the former there is no other science so beautiful 
and so important, none which contains such clear and precise proofs, 
theorems of such impeccable rigor, as the theory of numbers. Gauss 
said "If mathematics is the queen of sciences, then the theory of numbers 
is the queen of mathematics" . On the other hand, theory of numbers 
lies remote from those who are indifferent; they show little interest in 
its development, indeed they positively avoid it. The majority of 
students might, as regards their attitude, be put into the second class. 

I think that the reason for this remarkable division can be summarized 
as follows: On the one hand the theory of numbers is fundamental for 
all more thoroughgoing mathematical research] proceeding from entirely 
different fields, one comes at last, with extraordinary frequency, upon 
relatively simple arithmetic facts. On the other hand, however, the 
pure theory of numbers is an extremely abstract thing, and one does not 
often find the gift of ability to understand with pleasure anything so 
abstract. The fact that most textbooks are at pains to present the sub- 
ject in the most abstract way tends to accentuate this unattractiveness 
of the subject. I believe that the theory of numbers would be made more 
accessible, and would awaken more general interest, if it were presented 
in connection with graphical elements and appropriate figures. Although 
its theorems are logically independent of such aids, still one's compre- 
hension would be helped by them. I attempted to do this in my lectures 
in 1895/96 1 and a similar plan is followed by H. Minkowski in his book 
on Diophantische Approximationen 2 . My lectures were of a more ele- 
mentary introductory character, whereas Minkowski considers at an 
early point special problems in a detailed manner. 

As to textbooks in the theory of numbers, you will often find all you 
need in the textbooks in algebra. Among the large number of books 
on the theory of real numbers, I would mention especially Bachman's 
Grundlagen der neueren Zahlentheorie*. 

In the more special number -theoretic discussions which I shall give 
here, I shall keep touch with the points mentioned above and I shall 
endeavor especially to present the matter as graphically as possible, 
While I shall restrict myself to material that is valuable for the teacher, 
I shall by no means put it into a form suitable for immediate presentation 
to the pupils. The necessity for this arises from my experiences in 

1 Ausgew&hltes Kapitel der Zahlentheorie (mimeographed lectures written up 
by A. Sommerfeld and Ph. Furtwangler) . Second printing (already exhausted). 
Leipzig 1907. 

2 With an appendix: Fine Einfuhrung in die Zahlentheorie. Leipzig 1907- 
8 Sammlung Schubert No. 53. Leipzig 1907. [Second edition published by 

R. Hauszner 1921.] See also Carmichael, R. D., Theory of Numbers. Wiley. 

40 Arithmetic: Concerning Special Properties of Integers. 

examinations, which show me that the number-theoretic information of 
candidates is often confined to catchwords which have no thorough 
knowledge back of them. Every candidate can tell me that n is "trans- 
cendental" ; but many of them do not know what that means; I was 
told, once, that a transcendental number was neither rational nor ir- 
rational. Likewise I often find candidates who tell me that the number 
of primes is infinite, but who have no notion as to the proof, although 
it is so simple. 

I shall start my number-theoretic discussion with this proof, assuming 
that you are acquainted with the first two points metioned in our list. 
As a matter of history I remind you that this proof was handed on to 
us by Euclid, whose "elements" (Greek oroi%ia) contained not only 
his system of geometry, but also algebraic and arithmetic information in 
geometric language. Euclid's transmitted proof of the existence of in- 
finitely 'many prime numbers is as follows : Assuming that the sequence 
of prime numbers is finite, let it be 1 , 2, 3 , 5 , . . . , p\ then the number 
N = (1 2- 3 5 - p) 1 is not divisible by any of the numbers 
2 , 3 > 5 , . . p since there is always the remainder 1 ; hence N must 
either itself be a prime number or there are prime numbers larger than p . 
Either of these alternatives contradicts the hypothesis, and the proof 
is complete. 

In connection with the fourth point, the separation into prime factors, 
I should like to call to your attention one of the older factor tables: 
Chernac, Cribum Arithmeticum 1 , a large, meritorious work which de- 
serves, historically, all the more attention because it is so reliable. The 
name of the table suggests the sieve of Eratosthenes. The idea on which 
it was based is that we should discard gradually from the series of all 
integers those which are divisible by 2,3,5,-.., so that only the 
prime numbers would remain. Chernac gives the decomposition into 
prime factors of all integers up to 1020000 which are not divisible 
by 2, 3, or 5; all the prime numbers are marked with a bar. It 
was in the Chernac work that all the prime numbers lying within 
the limits stated above were first given. During the nineteenth century 
the determination was extended to all prime numbers as far as nine 

I turn now to the fifth point, the transformation of ordinary fractions 
into decimal fractions. For the complete theory I shall refer you to Weber- 
Wellstein, and I shall explain here only the principle of the method by 
means of a typical example. Let us consider the fraction \jp t where p 
is a prime number different from 2 and 5- We shall show that \jp is equal 
to an infinite periodic decimal, and that the number d of places in the 
period is the smallest exponent for which 10* 5 , when divided byp, leaves \ 

1 Deyenter 1811. 

Prime Numbers. 

as a remainder, or that, in the language of number theory, 6 is the 
smallest exponent which satisfies the "congruence" : 

The proof requires, in the first place, the knowledge that this congruence 
always has a solution. This is supplied by the theorem of Fermat, which 
states that for every prime number p except 2 and 5: 

We shall omit here the proof of this fundamental theorem, which is 
one of the permanent tools of every mathematician. Secondly, we must 
borrow from the theory of numbers the theorem that the smallest 
exponent in question, (3, is either p 1 itself or a divisor of p \ . We 

can apply this to the given value p and find that - is an integer N 
so that one has: 

-!5? = ^ + N 

P P^ 

If we now think of \tfjp, as well as l//>, converted into a decimal, 
the digits in the two decimals must be identical, since the difference 
is an integer. But since \tf/p is got from \\p by moving the decimal 
point 6 places to the right, it follows that the digits in the decimal 
expression of \/p are unaltered by this operation, in other words that 
the decimal fraction \lp consists of continued repetition of the same "period" 
of d digits. 

In order now to see that there cannot be a smaller period of 8' < <5 
digits one needs only to prove that the digit number 6' of every period 
must satisfy the congruence 10* 5 ' = 1 ; for we know that 6 was the 
smallest solution of this congruence. This proof will result if we pursue 
the preceding argument in the reverse direction. It follows from our 
assumption that 1/p and \tf'jp coincide in their decimal places, hence 

10^' 1 A' 

that --- - is an integer N', and therefore that 10 1 is divisible by p , 

or, in other words, that 10* 5 ' = 1 (mod^). This completes the proof. 
I will give you a few of the simplest instructive examples, which will 
show that d can take widely different values, both smaller than and 
equal to p 1 . Notice first that for: 

4 = 0.333... 

the number of digits in the period is 1 , and that in fact, 10 1 = 1 (mod 3)' 
Similarly we find 

^ = 0.0909..., 

whence d = 2, and correspondingly 10 1 slo,10 2 sl (mod 11). The 
maximum value = p 1 appears in the example: 

| -0,142857142857... . 

42 Arithmetic: Concerning Special Properties of Integers. 

Here <J = 6 and we have, in fact, 10 1 = 3 , 10 2 = 2, 10 3 = 6, 10 4 = 4, 
10 5 = 5 , and 10 6 = 1 (mod 7). 

Now let us take up, in a similar way, the sixth point of my list, 
continued fractions. I shall not present this, however, in the usual 
abstract arithmetic manner, since you will find it given elsewhere, e. g., 
in Weber-Wellstein. I shall take this opportunity to show you how 
number-theoretic things take on a clear and easily intelligible form 
through geometric and graphical presentation. In this use of geometric 
aids in number theory we are really only retracing the steps followed 
by Gauss and Dirichlet. It was the later mathematicians, say from i860 
on, who banished geometric methods from the theory of numbers. Of 
course, I can give here only the most important trains of thought and 
theorems, without proof, and I shall assume that you are not entire 
strangers to the elementary theory of continued fractions. My litho- 
graphed lectures on number theory 1 contain a thoroughgoing account. 

You know how the development of a given positive number co into a 
continued fraction arises. We separate out the largest positive integer n Q 
contained in co and write: 

a) = n + r , where ^ r Q < 1 , 

then, if r Q =f= 0, we treat l/r as we did co : 

l/^o = n \ + r i> where < ^ < 1 , 

and continue in the same way: 

\\r^ = n 2 + r 2 , where g r 2 < \, 
\jr z = n s + r 3 , where < r 3 < 1 , 

The process terminates after a finite number of steps if co is rational, 
because a vanishing remainder r v must appear in that case ; otherwise 
the process goes on indefinitely. In any case, we write, as the development 
of (o into a continued fraction : 

As an example, the continued fraction for n is 
^ = 3-14159265 =3 + - 


l + l 

292 +, 

1 See also Klein, F., Gesammelte Mathematische Abhandlungen, Vol. II, pp.209 
to 211. 

Continued Fractions. 


If we stop the development after the first, second, third, . . . partial 
denominator, we obtain rational fractions, called convergents: 

these give remarkably good approximations to the number co , or, to 
speak more exactly, each one of them gives an approximation which is 
closer than that given by any other rational fraction which does not have 
a larger denominator. Because of this property, continued fractions are 
of practical importance where one seeks the best possible approximation 
to an irrational number, or to a fraction with a large denominator (e. g. 
a many-place decimal) by means of a fraction having the smallest 
possible denominator. The following convergents of the continued frac- 
tion for a, converted into decimals, enable one to see how close the 
approximations are to the value n = 3,14159265 . . .' 

You will observe, moreover, in this example, that the convergents are 
alternately less than and greater than n. This is true in general, as is 
well known, that is the successive convergents of the continued fraction 
for oj are alternately less than and greater than co , and enclose it between 
ever narrowing limits. 

Let us now enliven these considerations with geometric pictures. 
Confining our attention to positive numbers, let us mark all those points 
in the positive quadrant of the %y plane (see Fig. 8) which have integral 
coordinates, forming thus a so called point lattice. Let us examine this 
lattice, I am tempted to say this "firmament" of points, with our point 
of view at the origin. The radius vector from to the point (x = a, 
y = b) has for its equation 

x _ a 
7 = T ' 

and conversely, there are upon every such ray, x/y = &, where A = a/b is 
rational, infinitely many integral points (ma, mb), where m is an arbi- 
trary whole number. Looking from 0, then, one sees points of the 
lattice in all rational directions and only in such directions. The field of 
view is everywhere "densely" but not completely and continuously filled 
with "stars". One might be inclined to compare this view with that 
of the milky way. With the exception of itself there is not a single 
integral point lying upon an irrational ray x/y = ft) , where ft) is irrational, 
which is very remarkable. If we recall Dedekind's definition of irrational 
number, it becomes obvious that such a ray makes a cut in the field 


Arithmetic: Concerning Special Properties of Integers. 

of integral points by separating the points into two point sets, one lying 
to the right of the ray and one to the left. If we inquire how these 
point sets converge toward our ray x/y = co , we shall find a very simple 
relation to the continued fraction for co . By marking each point (x = p v , 
y = q^ t corresponding to the convergent p v jq v , we see that the rays 
to these points approximate to the ray x]y = co better and better, alter- 
nately from the left and from the right, 
just as the numbers p v /q v approxi- 
mate to the number co. Moreover, if 
one makes use of the known number- 
theoretic properties of p v , q v , one 
finds the following theorem : Imagine 
pegs or needles affixed at all the integral 
points, and wrap a tightly drawn string 
about the sets of pegs to the right and to 
the left of the co-ray , then the vertices 
of the two convex string-polygons which 
bound our two point sets will be precisely 
the points (p v , q^ whose coordinates are 
the numerators and denominators of the 
successive conver gents to co , the left poly- 
gon having the even convergents, the 
Fig. 8. right one the odd. This gives a new, 

and, one may well say, an extremely 

graphic definition of a continued fraction. The representation in Fig. 8 
corresponds to the example 

CO = 


1 +1 

1 + 

which is the irrationality associated with the regular decagon. In this 
example, the first few vertices of the two polygons are 

left: ft = o, ? = 1 ; ft = 1 , 9 2 = 2; ft = 3, fc = 5; . . . 
right: p l = 1 , q l = 1 ; ft = 2, ? 3 = 3; ft = 5, ? 5 = 8; . . . 

The values p v , q v for n grow much more rapidly, so that one could 
hardly draw the corresponding representation. The proof of our theorem, 
which I cannot give here, can be found in detail on page 43 of in my 
lithographed lectures. 

I shall now pass on to the treatment of the seventh point, the Pytha- 
gorean numbers, where we shall use space perception in a somewhat 
different form. Instead of the equation: 

m 2 + 6 2 = c 2 . 

Pythagorean Numbers. 45 

whose integral solutions are sought, let us set: 

(2) */c = f, 6/c = ij 
and consider the equation: 

(3) P + i = 1 , 

with the problem of finding all the rational number-pairs , rj which 
satisfy it. Accordingly, we start from the representation of all rational 
points , r) (i.e. all points with rational coordinates f, rj), which will 
fill the ^-plane "densely". I 2 + *? 2 = 1 is the equation of the 
circle about the origin in this plane. It is our 
task to see how this circle threads its way through 
the dense set of rational points, in particular, to 
see which of these points it contains. We know a 
few such points of old, such as the intercepts 
with the axes, one of which, S ( = 1, q = 0) , 
we shall consider (see Fig. 9). All rays through 
5 are given by the equation 

(4) ? = *(* + !); Fig ' 9 - 

we call such a ray rational or irrational according as the parameter A is 
rational or not. We have now the double theorem that every rational 
point of the circle is projected from S by a rational ray and that every rational 
ray (4) meets the circle in a rational point. The first half of the theorem 
is obvious. We prove the second half by substituting from (4) in (3). 
This gives for the abscissas of the points of intersection the equation 


We know one solution of this equation, = 1 , which corresponds to 
the intersection S; for the other, one gets by easy calculation 

(5aj * = f^ 

and from (4) the corresponding ordinate 

(5b) ^ = TTF- 

From (5 a) and (5b) it follows that the second intersection is a rational 
point if A is rational. 

Our double theorem, now fully proved, can be stated also as follows. 
All the rational points of the circle are represented by formulas (5) -if A is 
an arbitrary rational number. This solves our problem and we need only 
to transform to whole numbers. For this purpose we put 

;. = n/m , 

46 Arithmetic: Concerning Special Properties of Integers. 

where n,m are integers and obtain from (5): 

_ m 2 n 2 _ 2m n 

f - m 2 + w a ' ^ - m 2 ~+ V 2 ' 

as the totality of rational solutions of (3). All integral solutions of the 
original equation (1), i.e., all Pythagorean numbers are therefore given by 
the equations 

a m 2 w 2 , b = 2 wn, c = m 2 + w 2 ; 

obtains the totality of solutions which have no common divisor if m 
and n take all pairs of relatively prime integral values. We have thus a 
graphic deduction of a result which usually appears very abstract. 
In this connection I should like to discuss the great Fermat. theorem. 
It is quite after the manner of the geometers of, antiquity that one 
should generalize the question regarding Pythagorean numbers, from 
the plane to space of three and more dimensions in the following manner. 
Is it possible that the sum of the cubes of two integers should be a cube? 
Or that the sum of two fourth powers should be a fourth power, etc.? 
In general, has the equation 

x n + y n = z n , 

where n is an arbitrary integer, solutions which are whole numbers'? To 
this question Fermat gave the answer no, in the theorem named after 
him : The equation % n + y n = z n has no integral solutions for integral values 
of n except when n = \ and n = 2 . Let me begin with a few historical 
notes. Fermat lived from 1601 to 1665 and was a parliamentary coun- 
cillor, i.e., a jurist, in Toulouse. He devoted himself, however, extensively 
and most fruitfully to mathematics so that he may counted as one of 
the greatest of mathematicians. Fermat' s name deserves a prominent 
place among those of the founders of analytic geometry, of infinitesimal 
calculus, and of the theory of probability. Of special significance 
however, are his attainments in the theory of numbers. All of his results 
in this field appear as marginal notes on his copy of Diophantus, the 
famous ancient master of number-theory who lived in Alexandria pro- 
bably about 300 A. D., i. e., about 600 years after Euclid. In this form 
they were published by his son five years after Fermat' s death. Fermat 
himself had published nothing, but he had, by means of voluminous 
correspondance with the most significant of his contemporaries, made 
his discoveries known, although only in part. It was in that edition 
of Diphantus that the famous theorem with which we are now concerned 
was found. Fermat wrote concerning it that "he had found a really 
wonderful proof, but the margin was too narrow to accommodate it" 1 . 
To this day, no one has succeeded in finding a proof of this theorem! 

1 See the edition issued by the Paris Academy: (Euvres de Fermat, vol. I, 
p. 291. Paris 1891, and vol. Ill, p. 241. Paris 1896. 

The Great Format Theorem. 


In order to orient ourselves somewhat as to its purport, let us 
inquire, as in the case of n = 2, in the first place about the rational 
solutions of the equation: 

^ n + r, n = 1 , 

i. e., about the relation of the curve which represents this equation to 
the totality of the rational points in the >j-plane. For n = 3 and n = 4 
the curves have approximately the appearance indicated in Fig. 10, 11 
They contain, at least, the points = o, ?7 1 and | = 1 , v\ = when 

Fig. 11. 

n = 3 , and the points = , >/ = 1 an d f = 1 , fl = when 
n = 4. The assertion of Fermat means, now, that these curves, unlike 
the circle considered above, thread through the dense set of the rational 
points without passing through a single one, except those just noted. 
The interest in this theorem rests on the fact that all efforts to find 
a complete proof of it have been, thus far, in vain. Among those who 
have attempted proof, one should, above all, mention Rummer, who 
advanced the problem materially by bringing it into relation with the 
theory of algebraic numbers, in particular with the theory of the n-th roots 


of unity (cyclotomic numbers). By using the n-th root of 1 , = e n , 
we can, indeed, separate z tl y n into n linear factors, and we may 
write the Fermat equation in the form 

*= (z -y)(z- ey) (z - &y) ... (z - ? n ~ l y) . 

The problem is therefore reduced to the separation of the n-th power 
of the integer % into n linear factors which shall be built up from two 
integers z and y and the number e, in the manner indicated. Kummer 
developed, for such numbers, theories quite similar to those which have 
long been known for the case of ordinary integers, theories, that is, 
which depend on the notions of divisibility and factorization. One 
speaks, accordingly, of integral algebraic numbers, and here, in particular, 
of cyclotomic numbers, because of the relation of the number e to the 
division of the circle. Fermat' s theorem is, then, for Kummer, a theorem 
on factorization in the domain of algebraic cyclotomic numbers. From this 

48 Arithmetic: Concerning Special Properties of Integers. 

theory he tried to deduce a proof of the theorem. He succeeded, in fact, 
for a very large number of values of n t for example for all values -of 'ft 
below 100. Among the larger numbers, however, there appeared ex- 
ceptional values for which no proof has been found, either by him or 
by the later mathematicians who continued his investigations. 

I must content myself with these remarks. You will find particulars 
concerning the state of the problem, and concerning Rummer's publica- 
tions in the Encyclopedia, Vol. I 2 , p. 714, at the end of the report by 
Hilbert, Theorie der Algebraischen Zahlkorper. Hilbert himself is among 
those who have continued and extended the investigations of Kummer 1 . 

It can indeed hardly be assumed that Fermat' s "wonderful proof" 
lay in this direction. For it is not very likely that he could have operated 
with algebraic numbers at a time when one was not even certain about 
the meaning of the imaginary. At that time, also, the theory of numbers 
was quite undeveloped. It received at the hands of Fermat himself far- 
reaching stimulation. On the other hand, one cannot assume that a 
mathematician of Fermat' s rank made an error in his proof, although 
such errors have occurred with the greatest mathematicians. Thus we 
must indeed believe that he succeeded in his proof by virtue of an 
especially fortunate simple idea. But as we have not the slightest 
indication as to the direction in which one could search for that idea, 
we shall probably expect a complete proof of Fefmat's theorem only through 
systematic extension of Rummer's work. 

These questions assumed new signifance when our Gottingen Science 
Association offered a prize of 100000 marks for the proof of Fermat* s 
theorem. This was a foundation of the mathematician Wolfskehl, who 
died in 1906. He had probably been interested all his life in Fermat' s 
theorem, and he bequeathed from his large fortune this sum for the 
fortunate person who should either establish the truth of the theorem 
of Fermat, or by means of a single example, exhibit its untruth 2 . Such 
a refutation would, be no simple matter, of course, because the theorem 
is already proved for exponents below 100 and one would have to start 
one's calculations with very large numbers. 

It will be clear, from my foregoing remarks, how difficult the winning 
of this prize must seem to the mathematician, who understands the 
situation and who knows what efforts have been made by Kummer 
and his successors to prove the theorem. But the great public thinks 

[ l A summarized account of the elementary investigations about Format's 
theorem is given in P. Bachmann, Das Fermatsche Problem. Berlin 191 9-] 

2 The detailed conditions governing competition for this prize (long since 
become valueless) were published in the Nachrichten d. Ges. d. Wissenschaften 
zu Gflttingen, business announcements 1908, p. 103 et seq., and copied into many 
other mathematical journals (Sec. e. g. Math. Ann. vol. 66, p. 143; Journal fur 
Mathematik, vol. 134, p. 313). 

Division of the Circle, 49 

otherwise. Since the summer of 1907, when the news of the prize was 
published in the papers (without authorization, by the way) we have 
received a prodigious heap of alleged "proof s". People of all walks of 
life, engineers, schoolteachers, clergymen, one banker, many women, 
have shared in these contributions. The common thing about them all 
is that they have no idea of the serious mathematical nature, of the problem. 
Moreover, they have made no attempt to inform themselves regarding 
it, but have trusted to finding the solution by a sudden flash of thought, 
with the inevitable result that their work is nonsense. One can see 
what absurdities are brought forth if one reads the numerous critical 
discussions of such proofs by A. Fleck (who is a practising physician 
by profession), Ph. Maennchen, and O. Perron, in Archiv fur Mathematik 
und Physik 1 . It is amusing to read these wholesale slaughterings, sad 
as it is that they are necessary. I should like to mention one example, 
which is related to our treatment of the case x 2 + y 2 = z*. The author 
seeks a rational parameter representation for the function x n + y n 
= z n (n > 2), and finds the result, long known from the theory of 
algebraic functions, that this, unlike the case n = 2, is not possible. 
Now this person overlooks the fact that a non-rational function can 
very well take on rational values for single 
rational values of the argument, and he x ' p ane 
therefore believes that he has proved the 
Fermat theorem. 

With this I close my remarks about 
Fermat' s theorem and come to the eighth 
point of my list, the problem of the division 
of the circle. I shall make use here of opera- 
tions with complex numbers, x + iy, as- 
suming that they are familiar to you, although Fig. 12. 

we shall consider them systematically later 

on. The problem is to divide the circle into n equal parts, or to construct 
a regular polygon of n sides. We identify the circle with the unit circle 
about the origin of the complex #y-plane and take x + iy = 1 as the 
first of the n points of division (see Fig. 12), in which n is chosen equal 
to five) ; then the n complex numbers belonging to the n vertices : 

. . fft . . . n n ,, _ . ., 

z = x + iy = cos -- h *sm - = e (k= 0, 1, ... , n 1) 

satisfy, according to De Moivre's theorem, the equation: 

z n = 1 , 

and with this the problem of the division of the circle is resolved into the 
solving of this simple algebraic equation. Since it has the rational root 

C 1 Vols. XIV, XV, XVI, XVII, XVIII (1901-19H).] 

Klein, Elementary Mathematics. 4 

50 Arithmetic: Concerning Special Properties of Integers. 

z = 1 , z n \ is divisible by z 1 , and there remains for the n 1 
other roots the so called cyclotomic equation 

z n-i + z n-2 + ... +Z 2 + z+ f = Q / 

an equation of degree n 1 , all of whose coefficients are + 1 . 

Since ancient times, interest has centered in the question as to what 
regular polygons can be constructed with ruler and compasses. It was 
known to the ancients that this construction was possible for the 
numbers n = 2 h , 3 , 5 (h an arbitrary integer), and likewise for the com- 
posite values n = 2 h 3 5 . Here the problem rested until the end of 
the eighteenth century when the young Gauss undertook its solution. 
He found the desired construction was possible with ruler and compasses 

for all prime numbers of the form p2^ 2 ' + \, but for no others. For 
the first values /i = 0, 1, 2, 3, 4 this formula yields, in fact, prime 
numbers, namely 

3, 5, 17, 257, 65537, 

of which the first two cases were already known, while the others were 
new. Of these the regular polygon of seventeen sides is especially famous. 
The fact that it can be constructed with ruler and compasses was first 
established by Gauss. Moreover, it is not known for what values of /u 
the above formula yields prime numbers. It has been known, for 
example, since Euler's time, that for [JL = 5 the number is composite. 
I shall not go farther into details, but rather outline the general con- 
ditions, and the significance of this discovery. You will find in Weber- 
Wellstein details concerning the regular polygon of seventeen sides. 

I should like to call to your attention especially the reprint of Gauss' 
diary in the fifty-seventh volume of the Mathematische Annalen (1903) 
and in Volume X, 1 (1917) of Gauss' Works. It is a small, insignificant 
looking book, which Gauss kept from 1796 on, beginning shortly before 
his nineteenth birthday. It was precisely the first entry which had to 
do with the possibility of constructing the polygon of seventeen sides 
(March 30, 1796); and it was this early important discovery which led 
Gauss to decide to devote himself to mathematics. The perusal of this 
diary is of the highest interest for every mathematician, since it permits 
one, farther on, to follow closely the genesis of Gauss' fundamental 
discoveries in the field of number theory, of elliptic functions, etc. 

The publication of that first great discovery of Gauss appeared as 
a short communication in the "Jenaer Literaturzeitung" of June 1, 1796, 
instigated by Gauss' teacher and patron, Hofrat Zimmermann, of Braun- 
schweig, and accompanied by a short personal note by the latter 1 . Gauss 
published the proof later in his fundamental number-theoretic work, 

1 Also reprinted in Mathematische Annalen, vol. 57, p. 6 (1903); and in Gauss' 
Works, vol. 10, p. 1 (1917). 

Division of the Circle. 51 

Disquisitiones Arithmeticae 1 in 1801; here one finds for the first time 
the negative part of the theorem, which was lacking in his communica- 
tion, that the construction with ruler and compasses is not possible for 

prime numbers other than those of the form 2 2 + 1 , e.g., for p = 7. I 
shall put before you here an example of this important proof of impossi- 
bilitythe more willingly because there is such a lack of understanding 
for proofs of this sort by the great public. By means of such proofs of 
impossibility modern mathematics has settled an entire series of famous 
problems, concerning the solution of which many mathematicians had 
striven in vain since ancient times. I shall mention, besides the con- 
struction of the polygon of seven sides, only the trisection of an angle 
and the quadrature of the circle with ruler and compasses. Nevertheless 
there are surprisingly many persons who devote themselves to these 
problems without having a glimmering of higher mathematics and 
without even knowing or understanding the nature of the proof of 
impossibility. According to their knowledge, which is mostly limited 
to elementary geometry, they make trials, by drawing, as a rule, auxiliary 
lines and circles, and multiply these finally in such number that no 
human being, without undue expenditure of time, can find his way out 
of the maze and show the author the error in his construction. A 
reference to the arithmetic proof of impossibility avails little with such 
persons, since they are amenable, at best, only to a direct consideration 
of their own "proof " and a direct demonstration of its falsity. Every 
year brings to every even moderately known mathematician a heap of 
such consignments, and you also, when you are at your posts, will get 
such proofs. It is well for you to be prepared in advance for such ex- 
periences and to know how to hold your ground. Perhaps it will be well 
for you, then, if you are master of a definite proof of impossibility in 
its simplest form. 

Accordingly, I should like to give you, in detail, the proof that it 
is impossible to construct the heptagon with ruler and compasses in the 
sense of geometry of precision. It is well known that every construction 
with ruler and compasses finds its arithmetic equivalent in a succession 
of square roots, placed one above another, and, conversely, that one 
can represent geometrically every such square root by the intersection 
of lines and circles. This you can easily verify for yourselves. We can 
formulate our assertion analytically, then, by saying that the equation 
of degree six 

2 6 + * 5 + 2 4 + 2 3 + Z 2 + Z + \ = , 

which characterizes the regular heptagon, cannot be solved by a succession 
of square roots in finite number. Now this is a so-called reciprocal equation, 

1 Reprinted Works, vol. I. 


J2 Arithmetic: Concerning Special Properties of Integers. 

i. e., it has, for every root z t also \\z as a root. This becomes obvious 
if we write it in the form : 

(1) * 3 + * 2 + * + 1 + y + ? + 73 = 0. 

We can reduce by half the degree of such an equation, if we take 

as a new unknown. By easy calculation, we obtain for x the cubic 

(2) x 3 + x 2 2x \ = 0, 

and one sees at once that the equations (1) and (2) are, or are not, both 
solvable by square roots. Moreover, we can represent x geometrically 
in connection with the construction of the heptagon. For, if we consider 

the unit circle in the complex plane, we see easily that the following re- 

/^ _ 

lations are obvious. If one designates by (? = the central angle of 
the regular heptagon, and remembers that z = cos q> + i sin <p and 

= cos <p i sin <p are the two vertices of the heptagon nearest to 


% = \ , then x = z + = 2 cos <p (Fig. 13). Thus, if one knows x, one 


can at once construct the heptagon. 

We must now show that the cubic equation (2) cannot be solved by 

square roots. The proof falls into an arithmetic and an algebraic part. 

z . p l ane We shall start by showing that the equa- 

tion (2) is irreducible, i. e. that its left side 
cannot be separated into two factors 
whose coefficients are rational numbers. 
Let us assume that the equation is re- 
ducible. Then its left side must have a 
linear factor with rational coefficients, 
and hence it must vanish for a rational 
Fjg 13 number pfq t where p and q are integers 

without a common divisor. But that 

means that /> 3 + p 2 q 2pq 2 q* =s= 0, or that p*, and therefore p 
itself, is divisible by q. In the same way it follows that <? 3 , and hence q, 
must be divisible by p . Consequently p = ? and the equation (2) must 
have the root x = 1 . But inspection shows that this is not the case. 
The second part of the proof consists, in showing that an irreducible 
cubic equation with rational coefficients is not solvable by square roots. It 
is essentially algebraic in nature, but because of the connection I shall 
give it here. Let us make the assertion in positive form. If a cubic 
equation with rational coefficients A , B , C : 

(8) /(*) = x* + Ax* 

Division of the Circle. 


can be solved by square roots, it must have a rational root, i. e., it is reducible. 
For the existence of a rational root # is equivalent to the existence of 
a rational factor % oc of / (x) and thus to reducibility. It is most 
important that this proof be preceded by a classification of all expressions 
that can be built up with square roots, or, more precisely, of all expressions 
that can be built up with square roots and rational numbers, in finite number, 
by means of rational operations. A concrete example of such a number is 

where a, b, . . ., f are rational numbers. Of course we are talking only 
about square roots which cannot be extracted rationally. All others must 
be simplified. Every such expression is a rational function of a certain 
number of square roots. In our example there are three. We shall first 
consider a single such square root, whose radicand, however, may have 
a form as complicated as one pleases. By its "order" we shall understand 
the largest number of root signs which appear in it, one above another. In 
the preceding example, oc , the roots of the numerator have the orders 
2 and 1 , respectively, while that of the denominator has the order 3 - 
In the case of a general square root expression we examine the orders 
of the different "simple square root expressions" of the sort just discussed, 
out of which the general expression is rationally constructed, and we 
designate the largest among them as the order fi of the expression in question. 
In our example, /* = 3. Now several "simple square root expressions 1 ' 
of order jm might appear in our expression and we consider their number, 
n, the "number of terms" of order p, as a second characteristic. This 
number is thought of as so determined that no one of the n simple 
expressions of order p can be rationally expressed in terms of the others of 
order ft, or of lower order. For example, the expression of order 1 

y 2 + y~3 + y~6 

has 2, not 3 , as the "number of terms" since /6 = }2 V3 . The example 
oi given above has n = 1 . 

We have thus assigned to every square root expression two finite 
numbers // , n which we combine in the symbol (/n , n) as the "characteristic" 
or "rank" of the root expression. When two root expressions have different 
orders we assign a lower rank to the one of lower order', when the orders 
are the same, the lower number of terms determines the lower rank. 

Now let us suppose that a root x l of the cubic equation (8) is expres- 
sible by means of square roots; and, to be explicit, by means of an 
expression of rank (p,ri). Selecting one of the n terms ]/ R of rank fi , let x 
be written in the form 

54 Arithmetic: Concerning Special Properties of Integers. 

where &, 0, y, d contain at most n \ terms of order /A and where R 
is of order p \ . Here y d ]/R is certainly different from zero ; 
for y 8 y^R = would imply either 6 = 7 = 0, which is obviously 
impossible, or]TR=y : 6, i.e., /7? would be rationally expressible by 
means of the other (n 1) terms of order ,a, which appear in x, and 
hence it would be superfluous. Multiplying numerator and denominator 
by y (>1/R, we find 


r - L + v V K > 

where P, Q are rational functions of <x,fi,y,d, that is, they contain 
at most (n 1) terms of order //, and, besides, only those of order 
// 1 , so that they have at most the rank (ft, n 1). Substituting this 
value of x in (8), we get 

f(x l ] = (P + QfRf + A(P + <?]/tf) 2 + B(P + QfR) + C = 0, 
and when we remove parentheses we obtain a relation of the form 

where M, 2V are polynomials in P, Q, R, that is, rational functions of 

a, ft, y> 6, R.'IfN^ 0, we should have ]//? == -M/N, i.e., ]//?" would 
be expressible rationally in terms of #,/?, y, 6, 7?, that is, by means of 
the other (n 1) terms of order // and others of lower order. But 
that is impossible, as remarked above, according to the hypothesis. 
Thus it follows necessarily that N = and hence also M = . From 
this we may conclude, that 

** = P- QfR 

is also a root of the cubic equation (8). For a comparison with the last 
equations yields at once 

/(*,) =M - NfR = Q. 

The proof may now be finished very simply and surprisingly. If x. 3 is 
the third root of our cubic equation, we have 

*! + # 2 + #3 = A , 

and hence x 3 = A (x l + x 2 ) = A 2 P 

is of the same rank as P and therefore certainly of lower rank than x l . 
If x 3 is itself rational, our theorem is proved. If not, we can make 
it the starting point of the same series of deductions. It appears that, 
in the case of the other roots, the higher rank must have been an illusion, 
so that, in particular, one of them has, actually, lower rank than x 3 . If 
we keep this up, back and forth among the roots, we see, each time, 
that the rank is really lower than we had thought. We must, then, 
of necessity, come finally to a root with the order ^ = . This demon- 

Ordinary Complex Numbers. 55 

strates the existence of a rational root of the cubic equation. We cannot 
continue our procedure beyond this point. The two other roots must 

then be, either themselves rational, or else of the form P = Q]/R, 
where P, Q , R are rational numbers. Hence we have shown that f (x) 
separates into a quadratic and a linear rational factor and is therefore 
reducible. Every irreducible cubic equation, and in particular, our equation 
for the regular heptagon, is insoluble by means of square roots. The proof 
is therefore complete that the regular heptagon cannot be constructed with 
ruler and compasses. 

You observe how simply and obviously this proof proceeds, and 
how little knowledge it really presupposes. For all that, some of the 
steps, especially the explanation of the classification of square root ex- 
pressions, demand a certain measure of mathematical abstraction. 
Whether the proof is simple enough to convince one of those mathe- 
matical laymen, mentioned above, of the futility of his attemps at an 
elementary geometric proof, I do not presume to decide. Nevertheless 
one should try to explain the proof slowly and clearly to such a person. 

In conclusion, I shall mention some of the literature on the question 
of regular polygons together with some, on the broader question of 
geometric constructibility in general which we have touched upon on 
this occasion. First of all, there is again Weber-Wellstein I (Sections 1 7 
and 18 in the fourth edition). Next let me mention the souvenir booklet 
Vortrdge iiber ausgewdhlte Fragen der Elementargeometrie 1 * which I pre- 
pared in 1895, on the occasion of a gathering of teachers in Gottingen. 
I might mention, as a more detailed and comprehensive substitute for 
this little book (which is out of print) the German translation, Fragen 
der Elementargeometrie***, of a compilation by F. Enriques in Bologna, 
where you will find information on all allied questions. 

I leave now the discussion of number theory, reserving the last 
point, the transcendence of n , for the conclusion of this course of lectures, 
and turn, in the next chapter, to our final extension of the number system. 

IV. Complex Numbers. 
1. Ordinary Complex Numbers 

Let me give, as a preliminary, some historical facts. Imaginary 
numbers are said to have been used first, incidentally, to be sure, by 
Cardan in 1 545, in his solution of the cubic equation. As for the further 

1 Worked up by F. Tagert. Leipzig 1895- 

2 Teilll: Die geometrischen Aufgaben, ihre Losung und Losbarkeit. Deutsch 
von H. Fleischer. Leipzig 190?. [2. Aufl. 1923-] See also Young, J. W. A. t Mono- 
graphs on Topics in Modern Mathematics. 

* Translation by Beman and Smith: Famous Problem of Geometry. Ginn, 
reprinted by Stechert, New York. 
** Problems of Elementary Geometry. 

56 Arithmetic: Complex Numbers. 

development, we can make the same statement as in the case of negative 
numbers, that imaginary numbers made their own way into arithmetic 
calculation without the approval, and even against the desires of individual 
mathematicians, and obtained wider circulation only gradually and to 
the extent to which they showed themselves useful. Meanwhile the mathe- 
maticians were not altogether happy about it. Imaginary numbers 
long retained a somewhat mystic coloring, just as they have today for 

every pupil who hears for the first time about that remarkable i = V 1 . 
As evidence, I mention a very significant utterance by Leibniz in the 
year 1702, "Imaginary numbers are a fine and wonderful refuge of the 
divine spirit, almost an amphibian between being and non-being". In 
the eighteenth century, the notion involved was indeed by no means 
cleared up, although Euler, above all, recognized their fundamental 
significance for the theory of functions. In 1748 Euler set up that remark- 
able relation: 

e ix = cos# + isinx 

by means of which one recognizes the fundamental relationship among 
the kinds of functions which appear in elementary analysis. The 
nineteenth century finally brought the clear understanding of the nature 
of complex numbers. In the first place, we must emphasize here the 
geometric interpretation to which various investigators were led about 
the end of the century. It will suffice if I mention the man who certainly 
went deepest into the essence of the thing and who exercised the most 
lasting influence upon the public, namely Gauss. As his diary, men- 
tioned above, proves incontrovertibly, he was, in 1 797, already in full 
possession of that interpretation, although, to be sure, it was published 
very much later. The second achievement of the nineteenth century 
is the creation of a purely formal foundation for complex numbers, 
which reduces them to dependence upon real numbers. This originated 
with English mathematicians of the thirties, the details of which I 
shall omit here, but which you will find in Hankel's book, mentioned 

Let me now explain these two prevailing foundation methods. We 
shall take first the purely formal standpoint, from which the consistency 
of the rules of operation among themselves, rather than the meaning 
of the objects, guarantees the correctness of the concepts. According 
to this view, complex numbers are introduced in the following manner, 
which precludes every trace of the mysterious. 

1 . The complex number x + iy is the combination of two real numbers 
x,y, that is, a number-pair, concerning which one adopts the conven- 
tions which follow. 

2. Two complex numbers x + iy, x' + iy f are called equal when 
x = x' ,y = y'. 

Ordinary Complex Numbers. 57 

3. Addition and subtraction are defined by the relation 

(* + iy} (*' + iy'} = (* *'} + i(y y') . 

All the rules of addition follow from this, as is easily verified. The mono- 
tonic law alone loses its validity in its original form, since complex 
numbers, by their nature, do not have the same simple order in which 
natural or real numbers appear by virtue of their magnitude. For the 
sake of brevity I shall not discuss the modified form which this gives 
to the monotonic law. 

4. We stipulate that in multiplication one operates as with ordinary 
letters, except that one always puts i 2 = 1; in particular, that 

(x + iy) (x f + iy') = (**' - yy') + i(xy' + x'y). 

It is easy to see that, with this, all the laws of multiplication hold, with 
the exception of the monotonic law, which does not enter into consideration. 

5. Division is defined as the inverse of multiplication] in particular, 
we may easily verify that 

1 = _5_ __ j y 

x + iy x 2 + y 2 x 2 + y 2 ' 

This number always exists except for x y = 0, i.e., division by zero 
has the same exceptional place here as in the domain of real numbers. 

It follows from this that operations with complex numbers cannot 
lead to contradictions, since they depend exclusively upon real numbers 
and known operations with them. We shall 
assume here that these are devoid of contra- 

Besides this purely formal treatment, we 
should of course like to have a geometric, or 
otherwise visual, interpretation of complex 
numbers and of operations with them, in which 
we might see a graphical foundation of consi- 
stency. This is supplied by common geometric 

interpretation, which, as you all know and as 

we have already mentioned, looks upon the 
totality of points (x,y) of the plane in an pig. u. 

xy-coordinate system as representing the totality 

of complex numbers z = x -\- iy. The sum of two numbers z, a follows 
by means of the familiar parallelogram construction with the two 
corresponding points and the origin 0, while the product z a is 
obtained by constructing on the segment 02 a triangle similar to 
001, where 1 is the point (x = 1 , y = 0) (Fig. 14). In brief, addition 
z 1 == z + a is represented by a translation of the plane into itself, mul- 
tiplication z 1 = z ' a by a similarity transformation, i.e., by a turning 
and a stretching, the origin remaining fixed. From the order of the points 

Jg Arithmetic: Complex Numbers. 

in the plane, considered as representatives of complex numbers, one 
sees at once what takes the place here of the monotonic laws for real 
numbers. These suggestions will suffice, I hope, to recall the subject 
clearly to your memory. 

I must call to your attention the place in Gauss in which this founda- 
tion of complex numbers, by means of their geometric interpretation, 
is set out with full emphasis, since it was this which first exhibited the 
general importance of complex numbers. In the year 1831 Gauss' 
researches carried him into the theory especially of integral complex 
numbers a + ib, where a, b are real integers, in which he developed 
for the new numbers the theorems of ordinary number theory concerning 
prime factors, quadratic and biquadratic residues, etc. We mentioned 
such generalizations of number theory, in connection with our discussion 
of Fermat's theorem. In his own abstract 1 of this paper Gauss 
expresses himself concerning what he calls the "true metaphysics of 
imaginary numbers". For him, the right to operate with complex 
numbers is justified by the geometric interpretation which one gives 
to them and to the operations with them. Thus he takes by no means 
the formal standpoint. Moreover, these long, beautifully written ex- 
positions of Gauss are extremely well worth reading. I mention here, 
also, that Gauss proposes the clearer word "complex", instead of 
"imaginary", a name that has, in fact, been adopted. 

2. Higher Complex Numbers, especially Quaternions 

It has occurred to everyone who has worked seriously with complex 
numbers to ask if we cannot set up other, higher, complex numbers, 
with more ne wunits than the one i and if we cannot operate with them 
logically. Positive results in this direction were obtained about 1840 
by H. Grassmann, in Stettin, and W. R. Hamilton, in Dublin, indepen- 
dently of each other. We shall examine the invention of Hamilton, the 
calculus of quaternions, somewhat carefully later on. For the present 
let us look at the general problem. 

We can look upon the ordinary complex number x + iy as a linear 


% 1 + y i 

formed from two different "units" \ and i , by means of the real parameters 
x and y . Similarly, let us now imagine an arbitrary number, n , of units 
e , e 2 , . . . , e n all different from one another, and let us call the totatily 
of combinations of the form x = x& + x 2 e 2 + . . . , + x n e n a higher 
complex number system formed from them with n arbitrary real numbers 
x lt x 2 , . . ., x n . If there are given two such numbers, say x, defined 
above, and 

y = 

1 See Werke, vol. II. 

Higher Complex Numbers, especially Quaternions. 59 

it is nearly obvious that we should call them equal when, and only when, 
the coefficients of the individual units, the so called "components" of the 
number, are equal in pairs 

The definition of addition and siibtr action, which reduces these operations 
simply to the addition and subtraction of the components, 

* it y = (*i yi)*i + (* a y 8 )*2 + ...,+ (x n y n ) e n , 

is equally obvious. 

The matter is more difficult and more interesting in the case of 
multiplication. To start with, we shall proceed according to the general 
rule for multiplying letters, i.e., multiply each i-th term of x by every 
k-ih term of y (i , k = \ , 2 , . . . , n) . This gives : 

x-y= ^ *y**i**- 

(', *=1, .... n) 

In order that this expression should be a number in our system, one must 
have a rule which represents the products d e* as complex numbers 
of the system, i. e., as linear combinations of the units. Thus one must 
have n 2 equations of the form: 

Then we may say that the number 

*-y = Z I Z 

(1=1. ..., n)l(i, /--I, ..., n) 

will always belong to our complex number system. Each particular 
complex number system is characterized by the method of determining 
this rule for multiplication, i.e., by the table of the coefficients Cud. 

If one now defines division as the operation inverse to multiplication, 
it turns out that, under this general arrangement, division is not always 
uniquely possible, even when the divisor does not vanish. For, the 
determination of y from x y = z requires the solution of the n linear 
equations xiytCiu = Zi for the n unknowns y lf . . ., y n , and these 


would have either no solution, or infinitely many solutions, if their 
determinant happened to vanish. Moreover, all the zi may be zero 
even when not all the Xi or not all the y^ vanish, i.e., the product of two 
numbers can vanish without either factor being zero. It is only by a skillful 
special choice of the numbers dki that one can bring about accord here 
with the behavior of ordinary numbers. To be sure, a closer investigation 
shows, when n > 2 , that, to attain this, we must sacrifice one of the other 
rules of operation. We choose as the rule that fails to be satisfied, one 
which appears less important under the circumstances. 

60 Arithmetic: Complex Numbers. 

Let us now follow up these general explanations by a more detailed 
discussion of quaternions as the example which, by reason of its applica- 
tions in physics and mathematics, constitutes the most important higher 
complex number system. As the name indicates, these are four-term 
numbers (n = 4) ; as a sub-class, they include the three-term vectors, 
which are generally known today, and which are sometimes discussed 
in the schools. 

As the first of the four units with which we shall construct quaternions, 
we shall select the real unit \ , (as in the case of ordinary complex num- 
bers). We ordinarily denote the other three units, as did Hamilton, 
by i,j,k, so that the general from of the quaternion is 

p = d + ia + jb + kc, 

where a,b,c,d are real parameters, the coefficients of the quaternion. 
We call the first component, the one which is multiplied by 1, and 
which corresponds to the real part of the common complex number, 
the "scalar part" of the quaternion, the aggregate ai + bj + ck of the 
other three terms its "vector part" . 

The addition of quaternions follows from the preceding general 
remarks. I shall give an obvious geometric interpretation, which goes 
back to that interpretation of vectors which is familiar to you. We 
imagine the segment, corresponding to the vector part of p , and having 
the projections a,b,c on the coordinate axes, as loaded with a weight 
equal to the scalar part. Then addition of p and p' = d' + ia' + jb' + kc' 
is accomplished by constructing the resultant of the 
two segments, according to the well known parallelogram 
law of vector addition (see Fig. 15), and then loading it 
w jth the sum of the weights, for this would then in fact 
represent the quaternion: 

(1) 'p + p'=(d + d') + i(a + a')+j(b + b')+k(c + c'). 

We come first to specific properties of quaternions 
when we turn to multiplication. As we saw in the general 
Fig. 15. case, these properties must be implicit in the conventions 

adopted as to the, products of the units. To begin with, 
I shall indicate the quaternions to which Hamilton equated the 
sixteen products of two units each. As its symbol indicates, we shall 
operate with the first unit 1 as with the real number 1, so that: 

(2a) l a = l f i-l =! = , / 1 =!/ = /, -1=1 -k = k. 

As something essentially new, however, we agree that, for the squares of 

the other units: 

(2b) i a = / 2 = *=-!, 

Higher Complex Numbers, especially Quaternions. fi\ 

and for their binary products: 

(2c) jk = +i, ki = j, ij = +k 

whereas for the inverted position of the factors: 
(2d) kj = i t ik=j, ji = k. 

One is struck here by the fact that the commutative law for multiplication 
is not obeyed. This is the inconvenience in quaternions which one must 
accept in order to rescue the uniqueness of division, as well as the theorem 
that a product should vanish only when one of the factors vanishes. 
We shall show at once that not only this theorem but also all the other laws 
of addition and multiplication remain valid, with this one exception, in 
other words, that these simple agreements are very expedient. 
We construct, first, the product of two general quaternions 

p = d + ia + jb + kc and q = w + ix + jy + kz. 
Let us start from the equation 

q' = p - q = (d -\- ia + j b + kc) (w + ix + jy + kz) ; 

and let us multiply out term by term. In carrying out this multiplication, 
we must note the order in the case of the units i,j , k. We must follow 
the commutative law for products composed of the components a,b,c,d, 
and for products of components and one unit, we must replace the 
products of units in accordance with our multiplication table, and we 
must then collect the terms having the same unit. We must then 
collect the terms having the same unit. We then have 

q' = pq = w' + ix' + jy' + kz' = (dw ax by cz) 

+ i(aw + dx + bz cy) 


+ j(bw + dy + ex az) 

+ k (cw + dz + ay bx) . 

The components of the product quaternion are thus definite simple 
bilinear combinations of the components of the two factors. If we 
invert the order of the factors, the six underscored terms change their 
signs, so that q' p, in general, is different from p q, and the difference 
is more than a change of sign as was the case with the individual units. 
Although the commutative law fails for multiplication, the distribu- 
tive and associative laws hold without change. For, if we construct on 
the one hand p(q + q^ , on the other pq + pq l by multiplying out 
formally without replacing the products of the units, we must, of 
necessity, get identical results, and no change can be brought about 
by then using the multiplication table. Further, the associative law 
must hold in general, if it holds for the multiplication of the units. 


Arithmetic: Complex Numbers. 

But this follows at once from the multiplication table, as the following 
example shows: 

In fact, we have: 

(*;)* = 

i(jk) = i i == 1. 

We shall now take up division. It will suffice to show that for every 
quaternion p ~ d -\- ia -}- jb -\- kc there is a definite second one, q, such 

We shall denote q appropriately by \jp . Division in general can be 
reduced easily to this special case, as we shall show later. In order to 
determine q, let us put, in equation (3), 

q 9 ^ \ = \ + Q'i + o-j + 0- k, 

and obtain, by equating components, the following four equations for 
four unknown components x,y,z,w of q: 

dw ax by cz = \ 
aw + dx cy + bz = 
bw -\- ex + dy az = 
cw b% + ay + dz 0. 

The solvability of such a system of equations depends, as is well known, 
upon its determinant, which, in the case before us, is a skew symmetric 
determinant, in which all the elements of the principal diagonal are the 
same, and all the pairs of elements which are symmetrically placed with 
respect to that diagonal are equal and opposite in sign. According to 
the theory of determinants, such determinants are easily calculated; 
and we find 

d a 



a d 



b c 



c -b 



By direct calculation this result can be easily verified. The real elegance 
of Hamilton's conventions depends upon this result, that the determinant 
is a power of the sum of squares of the four components of p\ for it 
follows that the determinant is always different from zero except when 
a = b = c = d = 0. With this one self evident exception (p = 0), 
the equations are uniquely solvable and the reciprocal quaternion q is 
uniquely determined.. 

Higher Complex Numbers, especially Quaternions. gi 

The quantity 

T = ]/0 2 + b 2 + c 2 + d 2 

plays an important role in the theory, and is called the tensor of p. 
It is easy to show that these unique solutions are 

a b c d 

x ~Y 2 , y -y^ z ~~"Y2' w = ~jv 

so that we have as the final result 

1 1 d ia jb he 

~~ ~~" 

If we introduce the conjugate value of p , as in ordinary complex numbers : 

p =. d ia jb kc, 
we can write the last formula in the form 

1= i 

p T* 

p.p = T 2 - a 2 + b 2 + c 2 + d 2 . 

These formulas which are immediate generalizations of certain properties 
of ordinary complex numbers. Since p is also the number conjugate 
to p, it follows also that: 

so that the commutative law holds in this special case. 

The general problem of division can now be solved. For, from the 

it follows, by multiplication by !//>, that 


whereas the equation 

9 ' P = l'> 
which one gets by changing the order of the factors, has the solution 

This solution is different, in general, from the other. 

Now we must inquire whether there is a geometric interpretation of 
quaternions in which these operations, together with their laws, appear 
in a natural form. In order to arrive at it, we start with the special 
case in which both factors reduce to simple vector s t i.e., in which the 


Arithmetic: Complex Numbers. 

scalar parts w,d, are zero, 

The formula (3) for multiplication then 

(<L t b,c) 

q 1 = p q = (ia + jb + kc) (ix + jy + kz) 

= (ax + by.+ cz) + i(bz cy) + j(cx az) + k(ay bx), 

i. e., when each of two quaternions reduces to a vector, their product consists 
of a scalar and a vector part. We can easily bring these two parts into 
relation with the different kinds of vector multiplication which are in 
use. The notions of vector calculus, which is far more wide spread than 
quaternion calculus, go back to Grassmann, although 
the word vector is of English origin. The two kinds 
of vector product with which one usually operates 
are designated now, mostly, by inner (scalar) product 
ax + by + cz (i.e., the scalar part of the above 
quaternion product, except for the sign), and outer 
(vector) product i (bz cy)+j (ex az) + k (ay bx) , 
(i.e., the vector part of the quaternion product. We shall give a geo- 
metric interpretation of each part separately. 

Let us lay off both vectors (a , b , c) and (x , y , z) , as segments, from 
the origin (Fig. 16). They terminate in the points (a>b,c) and (x,y t z) 

respectively, and have the lengths l = ^a 2 + b* + c 2 and /' = }/ x 2 + y * + z 2 . 

If (p is the angle between these two segments, then, according to well 

known formulas of analytic geometry, 
which I do not need to develop here, 
the inner product is: 

ax + by + cz = I 


Fig. 16. 



and the outer product, on the other 
hand, is itself a vector, which, as is 
easily seen, is perpendicular to the 
plane of I and I' and has the length 
I I' sin??. 

It is essential now to decide as to 
the sense of the product vector, i.e., 
toward which side of the plane deter- 
mined by I and V one is to lay off 
this vector. This sense is different 
according to the coordinate system 

which one chooses. As you know, one can choose two rectangular co- 
ordinate systems which are not congruent , i.e., which cannot be made to 
coincide with one another, by holding, say, the y- and the 2-axis fixed 
and reversing the sense of the #-axis. These systems are then sym- 
metric to each other, like the right and the left hand (Fig. 1 7) . The distinction 
between them can be borne in mind by the following rule: In the one 

Fig. 17. 

Quaternion Multiplication Rotation and Expansion. 55 

system, the x, y, and z axis lie like the outstretched thumb, fore finger and 

middle finger, respectively, of the right hand] in the other, like the same 

fingers of the left hand. These two systems are used confusedly in the 

literature; different habits obtain in different countries, in different 

fields, and, finally, with different writers, or even with the same writer. 

Let us now examine the simplest case, where p = i, q = j , these being 

the unit lengths laid off on the x and y axis. Then, since '/ = , 

the outer vector product is the unit length laid off 

on the 2-axis. (See Fig. 18.) Now one can trans- \ 

form i and j continuously into two arbitrary vectors 

p and q so that k transforms continuously into the 

vector component of p q without going through 

zero. Consequently the first factor, the second factor, 

and the vector product must always lie, with respect to Fig. is. 

each other, like the x, y, and z-axis of the system of 

coordinates, i.e., right-handed (as in Fig. 18) or left-handed (as in Fig. 16), 

according to the choice of coordinate system. (In Germany, now, the choice 

indicated in Fig. 18 is customary.) 

I should like to add a few words concerning the much disputed 
question of notation in vector analysis. There are, namely, a great many 
different symbols used for each of the vector operations, and it has been 
impossible, thus far, to bring about a generally accepted notation. 
At the meeting of natural scientists at Kassel (1903) a commission was 
set up for this purpose. Its members, however, were not able even to 
come to a complete understanding among themselves. Since their 
intentions were good, however, each member was willing to meet the 
others part way, so that the only result was that about three new 
notations came into existence! My experience in such things inclines 
me to the belief that real agreement could be brought about only if 
important material interests stood behind it. It was only after such 
pressure that, in 1881, the uniform system of measures according to 
volts, amperes, and ohms was generally adopted in electrotechnics and 
afterward settled by public legislation, due to the fact that industry 
was in urgent need of such uniformity as a basis for all of its calculations. 
But there are no such strong material interests behind vector calculus, 
as yet, and hence one must agree, for better or worse, to let every 
mathematician cling to the notation which he finds the most convenient, 
or if he is dogmatically inclined the only correct one. 

3. Quaternion Multiplication Rotation and Expansion 

Before we proceed to the consideration of the geometric meaning 
of multiplication of general quaternions, let us consider the following 
question. Let us consider the product q' = p q of two quaternions p 
and q, and let us replace p and q by their conjugates p and q, that 

Klein, Elementary Mathematics. 5 

66 Arithmetic: Complex Numbers. 

is, let us change the signs of a,b,c,x,y,z. Then the scalar part of 
the product, as given in (3), p. 61, remains unchanged, and only those 
factors of i,j,k which are not underscored will change sign. On the 
other hand, if we also reverse the order of the factors p and q, the 
factors of i,j,k which are underscored will change sign. Hence the 
product q'=q'p is precisely the conjugate of the original product q*\ 
and we have 

q' = p-q, q' = q-p, 

where q' is the conjugate of q'. If we multiply these two equations 
together, we obtain 

In this equation the order of the factors is essential, since the com- 
mutative law does not hold. We may apply the associative law, however, 
and we may write 

q'-q' = p'(q-q) -p. 

Since we have, by .p. 63, 
we may write 

q q = x * + 3/2 

y 2 

The middle factor on the right is a scalar, and the commutative law 
does hold for multiplication of a scalar by a quaternion, since M p 
= Md + i(Ma) + j(MV) + k(Mc) = pM . Hence we have 

w'* + x'* + y' 2 + z'* = pp(w 2 + x* + y 2 + z 2 ), 
and, since p -p is the square of the tensor of p, we find 1 
(I) w' z + x'* + y' 2 + *' 2 = (d* + a* + b* + c 2 ) (w* + x* + y 2 + * 2 ), 

that is, the tensor of the product of two quaternions is equal to the product 
of the tensors of the factors. This formula can be obtained also by direct 
calculation, by taking the values of w', %' , y', z' from the formula for 
a product given on p. 61 . 

We shall now represent a quaternion as the segment joining the 
origin of a four-dimensional space to the point (x , y , z , w) , in a manner 
exactly analogous to the representation of a vector in three-dimensional 
space. It is no longer necessary to apologize for making use of four- 
dimensional space, as was the custom when I was a student. All of 
you are fully aware that no metaphysical meaning is intended, and that 
higher dimensional space is nothing more than a convenient mathematical 
expression which permits us to use terminology analogous to that of 

1 This formula, in all that is essential, occurs in Lagrange's works. 

Quaternion Multiplication Rotation and Expansion. 67 

actual space representation. If we regard p as a constant, that is, if 
we regard a,b,c,d as constants, the quaternion equation 

9' = P ' q 

represents a certain linear tranformation of the points (x t y,z,w) of 
the four-dimensional space into the points (x f , y' ', 2', w'), since the 
equation assigns to every four-dimensional vector q another vector q' 
linearly. The explicit equations for this transformation, i.e., the ex- 
pressions for x' 9 y',z', w' as linear functions oix,y,z,w, may be obtained 
by comparison of the coefficients of the product formula (3), p. 61. 
The tensor equation (I) shows that the distance of any point from the 
origin, ^x z + y 2 + z 2 + w 2 , is multiplied by the same constant factor 
T = l/> + & 2 + 'c*~+d*, for all points of the space. Finally, by 
p. 62, the determinant of the linear transformation is surely positive. 

It is shown in analytic geometry of three-dimensional space that 
if a linear transformation of the coordinates x,y,z is orthogonal (that 
is, if it carries the expression % 2 + y 2 + z 2 into itself), and if the deter- 
minant of the transformation is positive, the transformation represents 
a rotation about the origin. Conversely, any rotation can be obtained in 
this manner. If the linear transformation carries x 2 + y 2 + z z into 
the similar expression in x', y', z' multiplied by a constant factor T 2 , 
however, and if the determinant is positive, the transformation re- 
presents a rotation about the origin combined with an expansion in the 
ratio T about the origin, or, briefly, a rotation and expansion. 

The facts just mentioned for three-dimensional space may be ex- 
tended to four-dimensional space. We shall say that our transformation 
of four-dimensional space represents in precisely the same sense a 
rotation and expansion about the origin. It is easy to see, however, that 
in this case we do not obtain the most general rotation and expansion 
about the origin. For our transformation contains only four arbitrary 
constants, namely, the components a , b , c , d of p , whereas, as we shall 
show immediately, the most general rotation and expansion about the 
origin in the four-dimensional space K 4 contains seven arbitrary con- 
stants. Indeed, in order that the general linear transformation should 
be a rotation and expansion, we must have 

If we replace x',y',z',w' by linear integral functions of x,y,z>w t 
we obtain a quadratic form in four variables, which contains (4 5)/2 = 10 
terms. Equating coefficients, we obtain ten equations. Since T is still 
arbitrary, these reduce to nine equations among the sixteen coefficients 
of the transformation. Hence there remain seven arbitrary constants. 
It is remarkable that in spite of this the most general rotation and 
expansion can be obtained by quaternion multiplication. Let n = t> + i& 

68 Arithmetic: Complex Numbers. 

+ j'P + ky be another constant quaternion. Then we may show, just as 
before, that the transformation q' = q-n, which differs from the 
preceding one only in that the order is reversed, represents a rotation 
and expansion of jR 4 . Hence the combined transformation 

(II) q' = p-q-7i = (d + ia + jb + kc)-q-(d + ioc + jfi + ky) 

also represents such a rotation and expansion. This transformation 
contains only seven (not eight) arbitrary constants, for the trans- 
formation remains unchanged if we multiply a , b , c , d by any real 
number and divide & , ft , y , <5 by the same number. It is therefore 
plausible that this combined transformation represents the general 
rotation and expansion of four-dimensional space. This beautiful result 
is actually true, as was shown by Cayley. I shall restrict myself to the 
mention of the historical fact, in order not to be drawn into too great 
detail. The formula is given in Cay ley's paper on the homographic 
transformation of a surface of the second order into itself 1 , in 1854, and also 
in certain other papers of his 2 . 

This formula of Cayley' s has the great advantage that it enables 
us to grasp at once the combination of two rotations and expansions. 
Thus, if a second rotation and expansion be given by the equation 

q" = w" + ix" + jy" + kz" = p f -q' n' , 
where p' and n' are new given quaternions, we find, by (II), 

?" = P' ' (P <1 ' ?*) ri , 
whence, by the associative law, 

<f=(P'. P ).q.(n.n'} 

q" = r q Q 

where r = p' p and Q = n n' are definite new quaternions. We have 
therefore obtained an expression for the rotation and expansion that 
carries q into q" in precisely the old form, and we see that the multipliers 
which precede and follow q in the quaternion product are, respectively, 
the products of the corresponding multipliers of q in the separate trans- 
formations which were combined, the order of the factors being neces- 
sarily as shown in the formula. 

This four-dimensional representation may seem unsatisfactory, and 
there may be a desire for something more tangible which can be re- 
presented in ordinary three-dimensional space. We shall therefore 
show that we can obtain similar formulas for the similar three-dimensional 

1 Journal fur Mathematik, 185$. Reprinted in Cayley's Collected Papers, vol. 2, 
p. 133. Cambridge 1889. 

2 See, for example, Recherches ultMeures sur les determinants gauches, loc. cit., 
p. 214. 

Quaternion Multiplication Rotation and Expansion. 69 

operations by a simple specialization of the formulas just given. Indeed 
the importance of quaternion multiplication for ordinary physics and 
mechanics is based upon these very formulas. I have said "ordinary", 
because I do not desire at this point to explain those generalizations 
of these science for which the preceding formulas apply without any 
modification. These generalizations are more immediate, however, than 
you may suppose. The new developments of electrodynamics which 
are associated with the principle of relativity, are essentially nothing 
else than the logical use of rotations and expansions in a four-dimensional 
space. These ideas have been presented and enlarged upon recently 
by Minkowski 1 . 

Let us remain, however, in three-dimensional space. In such a space, 
a rotation and expansion carries a point (x, y, z) into a point (x f ', y r , z'} 
in such a way that 

*'2 + y'2 + j'2 = ^2(^2 + y2 + ^2) ^ 

where M denotes the ratio of expansion of every length. Since the 
general linear transformation of (x 9 y 9 z) into (x' 9 y' 9 z') contains nine 
coefficients, and since the left-hand side of the preceding equation, 
after the insertion of the values of x', y', z'> becomes a quadratic form 
in x , y , z with six terms, the comparison of coefficients in the preceding 
equation leads to six equations, which reduce to five if the value 
of M is supposed arbitrary. Therefore the nine original coefficients 
of the linear transformation, which are subject to these five conditions, 
are reduced to four arbitrary constants. (Compare p. 67.) If such a 
linear transformation has a positive determinant, it represents, as was 
stated on p. 67, a rotation of space about the origin, together with an 
expansion in the ratio 1/M. If the determinant is negative, however, 
the transformation represents a rotation and expansion, combined with 
a reflection, such as, for example, the reflection defined by the equations 
x = x', y = y', z = z'. Moreover, it can be shown that the deter- 
minant of the transformation must have one of the two values M 3 . 
In order to represent these relationships by means of quaternions, 
let us first reduce the variable quaternions q and q' to their vectorial 
parts : 

q' = ix' -f. jy' -f kz', q = ix + jy + kz, 

which we shall think of as the three-dimensional vectors joining the 
origin to the positions of the point before and after the transformation, 
respectively. We shall show that the general rotation and expansion 

1 Since this was written, an extensive literature on the special theory of 
relativity mentioned above has appeared. Let me mention here my address Vber 
die geometrischen Grundlagen der Lorentzgruppe, Jahresbericht der deutschen 
Mathematiker-Vereinigung, vol. 19 (1910), p. 299, reprinted in Klein's Gesammelte 
mathematische Abhandlungen, vol. l, p. 533. 

70 Arithmetic: Complex Numbers. 

of the three-dimensional space is given by the formula (II) if p and n 
have conjugate values, that is, if we write q' = p q p ; or, in expanded 

\ ix > + jy > + kz > 

I = (d + ia + j b + kc) (ix + jy + kz) (d - ia jb - kc). 

In order to prove this, we must show first that the scalar part of the 
product on the right vanishes; that is, that q' is indeed a vector. To do 
this, we first mutiply p by q according to the rule for quaternion 
multiplication, and we find 

q' = [ ax by cz + i (dz + bz cy) 

+j(dy + ex az) + k (dz + ay bx)] [d ia jb kc] . 

After another quaternion multiplication, we actually find the scalar 
part of q' to be zero, whereas we find for the components of the vector 
part the expressions 

x' = (d* + a 2 b 2 c 2 )x + 2(ab cd)y+ 2(ac + bd)z 

y' = 2(ba + cd)x + (d 2 + b 2 c 2 a 2 )y h 2(bc ad)z 

z' = 2(ca bd)x + 2 


That these formulas actually represent a rotation and expansion becomes 
evident if we write the tensor equation for (1), which, by (I), is 

x'z + y '2 + 2 '2 _ ( d 2 + a z + b 2 + c 2 ) (x 2 + y 2 + z 2 ) (d* + a 2 + b 2 + c 2 ) , 

% '2 JL y'2 + /2 = 7 '4 . (^2 + y 2 + ^2) f 

where T ]/d 2 + a 2 -f b 2 + c 2 denotes the tensor of p. Hence, our 
transformation is precisely a rotation and expansion (see p. 69), provided 
the determinant is positive; otherwise it is such a transformation 
combined with a reflection. In any case, the ratio of expansion is M = T 2 . 
As remarked above, the determinant must have one of the two values 
M 3 = T 6 . If we consider the transformation for all possible values 
of the parameters a , b , c , d which correspond to the same tensor value T, 
which must obviously be different from zero, we see that the determinant 
must always have the value +T Q if it has that value for any single 
system of values of a,b 9 c,d m , for the determinant is a continuous 
function of a, 6, c, d, and therefore it cannot suddenly change in value 
from +T 6 to T 6 without taking on intermediate values. One set 
of values for which the determinant is positive is a = b = c =0, d - T 9 
since, by (2), the value of the determinant for these values oia,b,c,d,is 

d 2 , 0, 
0, d 2 , 
0, 0, d 2 

Quaternion Multiplication Rotation and Expansion. J\ 

It follows that the sign is always positive, and hence (1) always re- 
presents actually a rotation and expansion. It is easy to write down 
a transformation which combines a reflection with a rotation and an ex- 
pansion, for we need only combine the preceding transformation with 
the reflection x f = x,y' = y,z r = z, which is equivalent to 
writing the quaternion equation q' = p q p . 

We shall now show that, conversely, every rotation and expansion 
may be written in the form (1), or in the equivalent form (2). In the 
first place, this formula contains the four arbitrary constants which, 
as we saw on p. 69, are' necessary for the general case. That we can 
actually obtain any desired value of the expansion-ratio M = T 2 , 
any desired position of the axis of rotation, and any desired angle of 
rotation, by a suitable choice of the four arbitrary constants, can be 
seen by means of the following formulas. Let f , rj , f denote the direction 
cosines of the axis of rotation, and let co denote the angle of rotation. 
We have, of course, the well known relation 

(3) I s + rf + C 2 = 1 . 

I shall now prove that a , b , c , d are given by the equations 

d = T cos ~ ; 

(4) 2 

rn ,. . 0) . 0) rn . .CO 

a = 1 sin-- , b 1 Y\ sin - , c = 1 f sin , 


which, by (3), obviously satisfy the condition 

d 2 + a 2 + b 2 + c 2 = T 2 . 

When these relations have been proved, we can evidently obtain the 
correct values of a,b,c,d for any given values of T, , ?j, , co. 

To prove the relations (4), let us remark first that if a,b,c,d are 
given, the quantities co, ,??, are determined, and in such a way 
that (3) is satisfied. For, squaring and adding the equations (4), since T 
is the tensor of the quaternion p = d + ia + jb + kc, we have 

whence we see that (3) holds. It follows that , i? , C are fully determined 
by the relations 

which appear directly from (4). These equations express the fact that 
the point (a , b , c) lies on the axis of revolution of the transformation. 
This fact is easy to verify, for if we put x = a, y = b, z = c in (2), 

we find 

x' = (d* + a 2 + b 2 + c 2 ) a = T 2 a, 

y 9 = (d* + a 2 + Z> 2 + c 2 ) b = T 2 6, 
z' == (d 2 + a 2 + b 2 -f c 2 ) c = T 2 - c 

72 Arithmetic; Complex Numbers. 

that is, the point (a,b, c) remains on the same ray through the origin, 
which identifies it as a point on the axis of revolution. It remains 
only to prove that the angle co defined by (4) is actually the angle of 
rotation. This demonstration requires extended discussion which 
I can avoid now by remarking that the transformation (2) for T = 1 
reduces precisely to the transformation given by Euler for the revolution 
of the axes through the angle co about an axis of revolution whose 
direction cosines are , ??, t. This is to be found, for example, in Klein- 
Sommerfeld, Theorie des Kreisels, volume 1 x , where explicit mention 
of the theory of quaternions is given, or in Baltzer, Theorie und An- 
wendung der Determinanten*. 

Finally, if we substitute the values given by (4) in the equation (1), 
we obtain the very brief and convenient equation in quaternion form 
for the revolution through an angle co about an axis whose direction 
cosines are ,??,, combined with an expansion of ratio T 2 : 

ix' + jy f + kz' = T 2 {cos| + sin ~ (if + jr t + *)}{** + jy + kz} 

{CO . CD ,..,. , 7 f.\ 1 

cos - - sin -- (t f + 7 r/ + A) j . 

This formula expresses in a form that is easy to remember Euler' s 
formulas for rotation: the multipliers which precede and follow the 
vector ix + jy + kz, are, respectively, the two conjugate quaternions 
whose tensor is unity (so-called versor, that is, "rotator", in contra- 
distinction to tensor, " stretcher"), and then the whole result is to be 
multiplied by a scalar factor which is the expansion-ratio. 

We shall proceed now to show that when we specialize these formulas 
still further to two-dimensions, they become the well known formulas 
for the representation of a rotation and expansion of the xy plane by 
means of the multiplication of two complex numbers. (See p. 57.) 
For this purpose, let us choose the axis of rotation as the z axis 
( = r\ = 0, C = 1). Then the formula (5), for z = z' = 0, may be 
written in the form 


ix + jy = T 2 (cos ~ + k sin-^J (ix + jy) (cos -| Asin-^J, 

or, upon multiplication with due regard to the rules for products of the 

iod + jy = T 2 |cos-(;* + jy) + sm~(jx iy)||cos ~ fcsinyj 

.= r 2 |cos 2 ~-(ix-\-jy) + 2sin-^-cos-^-(/# iy) sin 2 -^- (ix + jy)\ 

= T*{(ix + jy) cosco + (joe iy)smo)} 
= T 2 (cosco + ksmco)(ix +jy). 

1 Leipzig 1897; 2nd printing, 1914. 2 Fifth edition, Leipzig 1881. 

Quaternion Multiplication Rotation and Expansion. 71 

If we now multiply both sides by the right-hand factor ( i), we obtain 

x' -f- ky' = T 2 (cosco + ksina)) (x + ky), 

which is precisely the rule for multiplying two ordinary complex numbers, 
and which can be interpreted as a rotation through an angle a) , together 
with an expansion in the ratio T 2 , except that we have used the letter k 

in place of the usual letter i to denote the imaginary unit ]/ 1. 

Let us now return to three-dimensional space, and let us modify 
the formula (1) so that it shall represent a pure rotation without an 
expansion. To do so, we must replace x', y' f z' by x' T 2 , y' T 2 , z r T 2 , 
that is, we must replace q' by q' T 2 . If we notice that p~^ = \lp ~p/T 2 , 
we may write the formula for a pure rotation in the form 

(6) ix' + jy' + kz' = p (ix + jy + kz) p-\ 

There is no loss of generality if we assume that p is a quaternion whose 
tensor is unity, that is, 

p = cos ~- + sin (iS + p? + *f), where | 2 + rf + 2 = \ , 
^ 2, 

whence we see that (6) results from (5) if T is set equal to unity. The 
formula was first stated in this form by Cayley in 1845 1 . 

We may express the composition of two rotations in a particularly 
simple form, precisely as we did above for four-dimensional space. 
Given a second rotation 

*x" + jy" + kz" = p' (ix' + jy' + kz') p'~\ 

P' = cos- + sin^ (r + jif + k?) 

the direction cosines of the axis of rotation being ', rj 1 ', f ', and the 
angle of rotation being a/, we may write 

ix" + jy" + kz" =p'-p- (ix + jy + kz)-p~ l - p'~ l 

as the equation for the resultant rotation. Hence the direction cosines 
of the axis or rotation, I", r\ n ', C", and the angle of rotation, co", for 
the resultant rotation, are given by the equation 

0" = cos ^ + sin^- (*" + iff' + k?') =p'-p. 

We have therefore found a brief and simple expression for the com- 
position of two rotations about the origin, whereas the ordinary formulas 
for expressing the resultant rotation appear rather complicated. Since 
any quaternion may be expressed as the product of a real number 

1 On certain results relating to quaternions, Collected Mathematical Papers, 
vol. 1 (1889), p. 123. According to Cayley's own statement (vol. 1, p. 586), however, 
Hamilton had discovered the same formula independently. 

74 Arithmetic: Complex Numbers. 

(its tensor) and the versor of a rotation, we have also found a simple 
geometric interpretation of quaternion multiplication as the com- 
position of the rotations. The fact that quaternion multiplication is 
not commutative then corresponds to the well known fact that the 
order of two rotations about a point cannot be interchanged, in general, 
without changing the result. 

If you desire to make a study of the historical development of 
the representations and applications of quaternions which we have 
discussed, I would recommend to you an extremely valuable report 
on dynamics written by Cayley himself: Report on the progress of the 
solution of certain special problems of dynamics*. 

I shall close with certain general remarks on the value and the 
dissemination of quaternions. For such a purpose, one should distinguish 
between the general quaternion calculus and the simple rule for 
quaternion multiplication. The latter, at least, is certainly of very 
great usefulness, as appears sufficiently from the preceding discussion. 
The general quaternion calculus, on the other hand, as Hamilton 
conceived it, embraced addition, multiplication, and division of 
quaternions, carried to an arbitrary number of steps. Thus Hamilton 
studied the algebra of quaternions; and, since he investigated also 
infinite processes, he may be said to have created a quaternion theory 
of functions. Since the commutative law does not hold, such a theory 
takes on a totally different aspect from the theory of ordinary complex 
variables. It is just to say, however, that these general and far-reaching 
ideas of Hamilton have not justified themselves, for there have not 
arisen any vital relationships and interdependencies with other branches 
of mathematics and its applications. For this reason, the general theory 
has aroused little general interest. 

It is in mathematics, however, as it is in other human affairs: there 
are those whose views are calmly objective; but there are always some 
who form regrettable personal prejudices. Thus the theory of quaternions 
has enthusiastic supporters and bitter opponents. The supporters, who 
are to be found chiefly in England and in America, adopted in 1907 
the modern plan by founding an "Association for the Promotion of the 
Study of Quaternions* ' . This organization was established as a thoroughly 
international institution by the Japanese mathematician Kimura, who 
had studied in America. Sir Robert Ball was for some time its president. 
They foresaw great, possible developments of mathematics to be secured 
through intensive study of quaternions. On the other hand, there are 
those who refuse to listen to anything about quaternions, and who go 
so far as to refuse to consider the very useful idea of quaternion mul- 

1 Report of the British Association for the Advancement of Science, 1862; 
reprinted in Cayley's Collected Mathematical Papers, Cambridge, vol. 4 (1891), 
pp. 552ff. 

Complex Numbers in School Instruction. 75 

tiplication. According to the view of such persons, all computation 
with quaternions amounts to nothing but computation with the four 
components; the units and the multiplication table appear to them to 
be superfluous luxuries. Between these two extremes, there are many 
who hold that we should always distinguish carefully between scalars 
and vectors. 

4. Complex Numbers in School Instruction 

I shall now leave the theory of quaternions and close this chapter 
with some remarks about the role which these concepts play in the cur- 
riculum of the schools. No one would ever think of bringing up 
quaternions in a secondary school, but the common complex numbers 
% + iy always come up for discussion. Perhaps it will be more interesting 
if, instead of telling you at length how it is done and how it ought to 
be done, I exhibit to you, by means of three books from different periods, 
how instruction has developed historically. 

I put before you, first, a book by Kastner who had a leading position 
in Gottingen in the second half of the eighteenth century. In those 
days one still studied, at the university, those elementary mathematical 
things which later, in the thirties of the nineteenth century, went over 
to the schools. Accordingly, Kastner also gave lectures on elementary 
mathematics, which were heard by large numbers of non-mathematical 
students. His book, which formed the basis of these lectures, was called 
Mathematische Anfangsgrunde*. The portion which interests us here 
is the second division of the third part: Anfangsgrunde der Analysis 
endlicher Grofien** 1 . The treatment of imaginary quantities begins there 
on page 20 in something like the following words: "Whoever demands 
the extraction of an even root of a 'denied' quantity (one said 'denied', 
then, instead of 'negative'), demands an impossibility, for there is no 
'denied' quantity which would be such a power". This is, in fact, quite 
correct. But on page 34 one finds: "Such roots are called impossible 
or imaginary", and, without much investigation as to justification, one 
proceeds quietly to operate with them as with ordinary numbers, 
notwithstanding their existence has just been disputed as though, so 
to speak, the meaningless became suddenly usable through receiving 
a name. You recognize here a reflex of Leibniz's point of view, according 
to which, imaginary numbers were really something quite foolish but 
they led, nevertheless, in some incomprehensible way, to useful results. 

Kastner was, moreover, a stimulating writer; he achieved quite 
a place in the literature as a coiner of epigrams. To cite only one of many 
examples, he expatiates, in the introduction of this book mentioned 

1 Third edition. Gottingen 1794. 
* Elements of Mathematics. 
** Elemements of Analysis of Finite Quantities. 

76 Arithmetic: Complex Numbers. 

above, on the origin of the word algebra, which, indeed, as the article 
"al" indicates, comes from the Arabic. According to Kastner, an 
algebraist is a man who "makes" fractions "whole", who, that is, treats 
rational functions and reduces them to a common denominator, etc. 
It is said to have referred, originally, to the practice of a surgeon in 
mending broken bones. Kastner then cites Don Quixote, who went to 
an algebraist to get his broken ribs set. Of course, I shall leave undecided, 
whether Cervantes really adopted this form of expression or whether 
this is only a lampoon. 

The second work which I put before you is more recent, by a whole 
series of years, and comes from the Berlin professor M. Ohm: Versuch 
eines vollstandig konsequenten Systems der Mathematik* 1 ; a book with 
purpose similar to that of Kastner and at one time widely used. But 
Ohm is much nearer the modern point of view, in that he speaks clearly 
of the principle of the extension of the number system. He says, for 
example, that, just like negative numbers, so j/ 1 must be added to 
the real numbers as a new thing. But even his book lacks a geometric 
interpretation, since it appeared before the epoch-making publication 
by Gauss (1831). 

Finally, I lay before you, out of the long list of modern school books, 
one that is widely used: Bardeys Aufgabensammlung 2 . The principle 
of extension comes to the fore here, and, in due course, the geometric 
interpretation is explained. This may be taken as the general position 
of school instruction today, even if , at isolated places, the development 
has remained at the earlier level. The point of view adopted in this 
book seems to me to yield the treatment best adapted to the schools. 
Withhout tiring the pupil with a systematic development, and without, 
of course, going into logically abstract explanations, one should explain 
complex numbers as an extension of the familiar number concept, and 
should avoid any touch of mystery. Above all, one should accustom 
the pupil, at once, to the graphic geometric interpretation in the complex 

With this, we come to the end of the first main part of the course, 
which was dedicated to arithmetic. Before going over to similar dis- 
cussions of algebra and analysis, I should like to insert a somewhat 
extended historical appendix in order to throw new light upon the 
general conduct of instruction at present, and upon those features of it 
which we would improve. 

1 Nine volumes. Berlin 1828. Vol.1: Arithmetik und Algebra, p. 276. 

* An Attempt to Construct a Consistent System of Mathematics. 

[ 2 See also the Reformausgabe of Bardeys Aufgabensammlung, revised by 
W. Lietzmann and P. Ziilke. Oberstufe. Verlag Teubner. Leipzig.] See also 
Fine, H., The Number-System in Algebra. Heath. Fine, H., College Algebra. 

Development and Structure of Analysis. 77 

Concerning the Modern Development and the General 
Structure of Mathematics 

Let me proceed from the remark that, in the history of the development 
of mathematics up to the present time, we can distinguish clearly two 
different processes of growth, which now change places, now run side 
by side independent of one another, now finally mingle. It is difficult 
to put into vivid language the difference which I have in mind, because 
none of the current divisions fits the case. You will, however, under- 
stand from a concrete example, namely, if I show how one would compile 
the elementary chapters of the system of analysis in the sense of each of 
these two processes of development. 

If we follow the one process, which we will call briefly Plan A, 
the following system presents itself, the one which is most widespread 
in the schools and in elementary textbooks. 

1. At the head stands the formal theory of equations, that is to say, 
the operating with rational integral functions and the handling of the 
cases in which algebraic equations can be solved by radicals. 

2. The systematic pursuit of the idea of power and its inverses yields 
logarithms, which prove to be so useful in numerical calculation. 

3. Whereas (up to this point) the analytic development is kept quite 
separate from geometry, one now borrows from this field, which yields 
the definitions of a second kind of transcendental functions, the trigono- 
metric functions, the further theory of which is built up as a new separate 

4. Then follows the so called "algebraic analysis' ', which teaches 
the development of the simplest functions into infinite series. One considers 
the general binomial, the logarithm and its inverse, the exponential func- 
tion, together with the trigonometric functions. Similarly, the general 
theory of infinite series and of operations with them belongs here. It is 
here that the surprising relations between the elementary transcendentals 
appear, in. particular the famous Euler formula 

e ix = cosx + isin x. 

Such relations seem the more remarkable because the functions which 
occur in them had been originally defined in entirely separate fields. 

5. The consistent continuation, beyond the schools, of this structure, 
is the Weierstrass theory of functions of a complex variable, which 
begins with the properties of power series. 

Let us now set over against this, in condensed form the second 
process of development, which I shall call Plan B. Here the controlling 
thought is that of analytic geometry, which seeks a fusion of the perception 
of number with that of space. 

1. We begin with the graphical representation of the simplest functions, 
of polynomials, and rational functions of one variable. The point in 

78 Modern Development and Structure of Mathematics. 

which the curves so obtained meet the axis of abscissas put in evidence 
the zeros of the polynomials, and this leads naturally to the theory of the 
approximate numerical solution of equations. 

2. The geometric picture of the curve supplies naturally the intuitive 
source both for the idea of the differential quotient and that of the integral. 
One is led to the former by the slope of the curve, to the latter by the 
area which is bounded by the curve and the axis of abscissas. 

3. In all those cases in which the integration process (or the process 
of quadrature, in the proper sense of that word) cannot be carried out 
explicitly with rational and algebraic functions, the process itself gives 
rise to new functions, which are thus introduced in a thoroughly natural 
and uniform manner. Thus, the quadrature of the hyperbola defines the 

"*dx f 

while the quadrature of the circle can easily be reduced to the integral 




= arcsm#, 

that is, to the inverses of the trigonometric functions. You know that 
the same line of thought, pursued farther, leads to new classes of 
functions of higher order, in particular to elliptic functions. 

4. The development into infinite power series of the functions thus 
introduced is obtained by means of a uniform principle, namely Taylor's 

5. This method carried higher, yields the Cauchy-Riemann theory of 
analytic functions of a complex variable, which is built upon the Cauchy- 
Riemann differential equations and the Cauchy integral theorem. If we 
try to put the result of this survey into definite words, we might say 
that Plan A is based upon a more particularistic conception of science 
which divides the total field into a series of mutually separated parts and 
attempts to develop each part for itself, with a minimum of resources and 
with all possible avoidance of borrowing from neighboring fields. Its ideal 
is to crystallize out each of the partial fields into a logically closed system. 
On the contrary, the supporter of Plan B lays the chief stress upon the 
organic combination of the partial fields, and upon the stimulation which 
these exert one upon another. He prefers, therefore, the methods which open 
for him an understanding of several fields under a uniform point of view. 
His ideal is the comprehension of the sum total of mathematical science 
as a great connected whole. 

One cannot well be in doubt as to which of these two methods has 
more life in it, as to which would grip the pupil more, in so far as he is 
not endowed with a specific abstract mathematical gift. In order to 
bring this home, think only of the example of the functions e x and sin x, 

Development of Analysis in the Schools. 79 

about which we shall later have much to say along just this line! In 
Plan A, which the schools, unfortunately, follow almost exclusively, 
both functions come up in thoroughly heterogeneous fashion: e x or, 
as the case may be, the logarithm, is introduced as a convenient aid in 
numerical calculation, but sin x appears in the geometry of the triangle. 
How can one understand, thus, that the two are so simply connected, 
and, more, that the two appear again and again in the most widely 
differing fields which have not the least to do, either with the technique 
of numerical calculation or with geometry, and always of their own 
accord, as the natural expression of the laws that govern the subject 
under discussion ? How far these possibilities of application go is shown 
by the names compound interest law or law of organic growth, which have 
been applied to e x , and likewise by the fact that sin x plays a central 
role wherever one has to do with vibrations. But in Plan B these 
connections make their appearance quite intelligibly, and in accord with 
the significance of the functions, which is emphasized from the start. The 
functions e x and sin x arise here, indeed, from the same source, the 
quadrature of simple curves, and one is soon led from there, as we shall 
see later on, to the differential equations of simplest type 

de x _ x 

-j e , T' J ~O 

dx dx* 

respectively, which lie naturally at the basis of all those applications. 

For a complete understanding of the development of mathematics 
we must, however, think of still a third Plan C, which, along side of 
and within the processes of development A and B, often plays an 
important role. It has to do with a method which one denotes by the 
word algorithm, derived from a mutilated form of the name of an Arabian 
mathematician. All ordered formal calculation is, at bottom, algorithmic, 
in particular, the calculation with letters is an algorithm. We have 
repeatedly emphasized what an important part in the development of 
the science has been played by the algorithmic process, as a quasi- 
independent, onward-driving force, inherent in the formulas, operating 
apart from the intention and insight of the mathematician, at the time, 
often indeed in opposition to them. In the beginnings of the infinitesmal 
calculus, as we shall see later on, the algorithm has often forced new 
notions and operations, even before one could justify their admissibility. 
Even at higher levels of the development, these algorithmic considera- 
tions can be, and actually have been, very fruitful, so that one can justly 
call them the groundwork of mathematical development. We must then 
completely ignore history, if, as is sometimes done today, we cast these 
circumstances contemptuously aside as mere "formal" developments. 

Let me now follow more carefully through the history of mathematics 
the contrast of these different directions of work, confining myself of course 

gO Modern Development and Structure of Mathematics. 

to the most important features of the development. The thoroughgoing 
difference between A and B, within the whole field of mathematics, will 
appear here more clearly than it did above, where our thoughts were 
directed only to analysis. 

With the ancient Greeks we find a sharp separation between pure and 
applied mathematics, which goes back to Plato and Aristotle. Above all, 
the well known Euclidean structure of geometry belongs to pure mathe- 
matics. In the applied field they developed, especially, numerical calcula- 
tion, the so called logistics (Aoyoc = general number, see p. 32). To 
be sure, the logistics was not highly regarded, and you know that this 
prejudice has, to a considerable extent, maintained itself to this day 
mainly, it is true, with only those persons who themselves cannot 
calculate numerically. The slight esteem for logistics may have been 
due in particular to its having been developed in connection with 
trigonometry and the needs of practical surveying, which to some does not 
seem sufficiently aristocratic. In spite of this fact, it may have been 
raised somewhat in general esteem by its application in astronomy, 
which, although related to geodesy, always has been considered one of 
the most aristocratic fields. You see, even with these few remarks, 
that the Greek cultivation of science, with its sharp separation of the 
different fields, each of which was represented with its rigid logical 
articulation, belonged entirely in the plan of development A . Nevertheless 
the Greeks were not entire strangers to reflections in the sense of Plan B, 
and these may have served them for heuristic purposes, and for a first 
communication of their discoveries, even if the form A appeared to 
them indispensable for the final presentation. This is indicated quite 
pointedly in the recently discovered manuscript of Archimedes^, in which 
he exhibits his calculations of volume through mechanical considerations, 
in a thoroughly modern, pleasing way, which has nothing to do with 
the rigid Euclidean system. 

Besides the Greeks, in ancient times, the Hindus, especially, played 
a mathematical role as creators of our modern system of numerals, and 
later the Arabs, as its transmitters. The first beginnings of operating with 
letters were made also by the Hindus. These advances belong obviously 
to the algorithmic course of development C. 

Coming now to modern times, we can, first of all, date the mathematical 
renaissance from about 1500, which produced an entire series of great 
discoveries. As an example, I mention the formal solution of the cubic 
equation (Cardan's formula), which was contained in the "Ars Magna" 
of Cardano, published in 1545, in Niirnberg. This was a most significant 
work, which holds the germs of the modern algebra, reaching out beyond 

1 Cf. Heiberg und Zeuthen, Eine neue Schrift des Archimedes. Leipzig 190?. 
Reprint from Bibliotheca Mathematica. Third series, vol. VIII. See also HEATH, 
T. L. t The Works of Archimedes. Cambridge University Press. 

Brief Survey of the History of Mathematics. 81 

the scheme of ancient mathematics. To be sure, this work is not Cardano's 
own, for he is said to have taken from other authors not merely his 
famous formula but other things as well. 

After 1550 trigonometric calculation was in the foreground. The first 
great trigonometric tables appeared in response to the needs of astronomy, 
in connection with which I will mention only the name of Copernicus. 
From about 1600 on, the invention of logarithms continued this develop- 
ment. The first logarithmic tables, which a Scotchman Napier (or Nep6r) 
compiled, contained, in fact, only the logarithms of trigonometric 
functions. Thus we see, during these hundred years,. a path of develop- 
ment which corresponds to the Plan A. 

We come now, in the seventeenth century, to the modern era proper, 
in which the Plan B comes distinctly into the foreground. In 1637 
appeared the analytic geometry of Descartes, which supplies the funda- 
mental connection between number and space for all that follows. A 
reprint 1 makes this work conveniently accessible. Now come, in close 
sequence with this, the two great problems of the seventeenth century, the 
problem of the tangent, and the problem of quadrature, in other words, 
the problems of differentiating and integrating. For the development of 
differential and integral calculus, in a proper sense, there was lacking 
only the knowledge that these two problems are closely connected, that 
one is the inverse of the other. A recognition of this fact was the principal 
item in the great advance which was made at the end of the seventeenth 

But before this, in the course of the century, the theory of infinite 
series, in particular, of power series, made its appearance, and not, in- 
deed, as an independent subject, in the sense of the algebraic analysis 
of today, but in closest connection with the problem of quadrature. Nicolaus 
Mercator (the German name "Kaufmann" latinized; 1620 1687), not 
the inventor of the Mercator projection, was a pioneer here. He had 
the keen idea of converting the fraction 1/(1 + x) into a series, by dividing 
out, and of integrating this series term by term, in order to get the series 
development for log (1 + x): 

That is the substance of his procedure, although he did not, of course, 
use our simple symbols f , dx, . . ., but rather a much more clumsy 
form of expression. In the sixties, Isaac Newton (1643 1727) took over 
this process, to apply it to the series for the general binomial, which he 
had set up. In this process he drew his conclusions by analogy, basing 

1 Descartes, R., La Gtomttrie. Nouvtlle Edition. Paris 1886. Translation 
by Smith, D. E., and Latham, M. L., 1925. Open Court. 

Klein, Elementary Mathematics. 6 

arc sm#. 

82 Modern Development and Structure of Mathematics. 

them on the known simplest cases, without having a rigorous proof 
and without knowing the limits within which the series development 
was valid. We observe here, again, the operation of the algorithmic 


process C. By applying the binomial series to J==L= (1 tf 2 )"" 1 / 2 

V i x 

C x dx 

and using Mercator's process, he gets the series for I f = 

Jo V 1 x 

By a very skillful inversion of this series, and also of the one for log x , 
he finds the series for sin x and for e x . The conclusion of this chain of 
discoveries is due to Brook Taylor (1685 1731) who, in 1715, published 
his general principle for developing functions into power series. 

As is indicated above, the origin of infinitesimal calculus, at the end 
of the seventeenth century, was due to G. W. Leibniz (16461716) and 
Newton. The fundamental idea with Newton is the notion of flowing. Both 
variables A;, y,aretought of as functions, <p(t), ip(t), of the time t\ and as 
the time "flows", they flow also. Newton, accordingly, calls the variable 
fluens and designates as fluxion x, y , that which we call differential 
coefficient. You see how everything here is based firmly on intuition. 

It was much the same with the representation of Leibniz, whose first 
publication appeared in 1684. He himself declares that his greatest 
discovery was the principle of continuity in all natural phenomena, that 
"Natura non facit saltum". He bases his mathematical developments 
upon this concept, another example of the Plan B. However, the 
influence of the algorithm C is very strong, also, with Leibniz. We get 
from him the algorithmically valuable symbols dy/dx and f f (x) dx. 

The sum total, however, of this cursory view is that the great discov- 
eries of the seventeenth century belong primarily to the plan of develop- 
ment B. 

In the eighteenth century, this period of discovery continues at first 
in the same direction. The most distinguished names to be mentioned 
here are L. Euler (1707-1783) and J. L. Lagrange (17361813). Thus 
the theory of differential equations, in the most general sense, including 
the calculus of variations, were developed, and analytical geometry and 
analytical mechanics were extended. Everywhere there was a gratifying 
advance, just as in geography, after the discovery of America, the new 
countries were first traversed and explored in all directions. But just 
as there was, as yet, no thought of exact surveys, just as at first one had 
entirely false notions as to the location of these new places (Columbus, 
indeed, thought at first, that he had reached Eastern Asia!), just so, 
in the newly conquered region of mathematics, that of infinitesimal 
calculus, one was, at first, far removed from a reliable logical orientation. 
Indeed one even cherished illusions concerning the relation of the calculus 
to the older familiar fields, in thatone looked upon infinitesimal calculus 
as something mystical that in no way submitted to a logical analysis. 

Brief Survey of the History of Mathematics. 8} 

Just how untrustworthy the ground was on which the theory stood, 
became manifest only when it was attempted to prepare textbooks which 
should present the new subject in an intelligible way. Then it became 
evident that the method of procedure B was no longer adequate, and it 
was Euler who first abandoned it. He had, to be sure, no serious doubts 
concerning infinitesimal calculus, but he thought that it caused too 
many difficulties and misgivings for the beginner. For this pedagogical 
reason he thought it advisable to give a preparatory course, such as 
he provided in his text book Introductio in analysin infinitorum (1748), 
and which we call today algebraic analysis. To this he relegated, in 
particular, the theory of infinite series and other infinite processes, which 
he then afterwards used as a foundation in constructing the infinitesimal 

Lagrange took a much more radical course, nearly fifty years later, 
in his Th&orie des fonctions analytiques, in 1797. He could satisfy his 
scruples as to the current foundations of infinitesimal calculus only by 
discarding it entirely, as a general branch of knowledge, and by consider- 
ing it as an aggregate of formal rules applying to certain special classes 
of functions. Indeed, he considers exclusively such functions as can be ex- 
pressed by means of power series: 

fix) = a + a^x + a 2 x 2 + a 3 x* + - - . 

He calls these analytic functions, meaning thereby functions which appear 
in analysis and with which one can reasonably hope to do something. 
The differential quotient of such a function, f (x), is then defined, purely 
formally , by means of a second power series, as we shall see later. Diffe- 
rential and integral calculus was concerned, then, with the mutual 
relations of power series. This restriction to formal consideration ob- 
viated, for a time, of course, a number of difficulties. 

As you see, the turn which Euler gave, and still more, the entire method 
of Lagrange, belongs strictly to the direction A, in that the perceptual genetic 
development is replaced by a rigorous closed circle of reasoning. These 
two investigators have had a profound influence upon instruction in the 
secondary schools, and when the schools today study infinite series, or 
solve equations by means of power series according to the so called 
method of indeterminate coefficients, but decline to take up differential 
and integral calculus proper, they are exhibiting precisely the after effect 
of Euler y s "introductio" and of Lagrange' s thought. 

The nineteenth century, to which we come now, begins primarily 
with a more secure foundation of higher analysis, by means of criteria of 
convergence, about which one had hitherto thought but little. The 
eighteenth century was the "blissful" period, during which one did 
not distinguish between good and bad, convergent and divergent. Even 
in Euler 's Introductio, divergent and convergent series appear peaceably 

84 Modern Development and Structure of Mathematics. 

side by side. But, at the beginning of the new century Gauss (17771855) 
and Abel (18021829) made the first rigorous investigations regarding 
convergence; and in the twenties Cauchy (17891857) developed, in 
lectures and in books, the first rigorous founding of infinitesimal calculus 
in the modern sense. He not only gives an exact definition of the differential 
quotient, and of the integral, by means of the limit of a finite quotient and 
of a finite sum, respectively, as had previously been done, at times; but, 
by means of the mean-vahie theorem he erects upon this, for the first 
time, a consistent structure for the infinitesimal calculus. We shall come 
back to this fully later on. These theories also partake of the nature 
of Plan A , since they work over the field in a logical sytematic way, 
quite apart from other branches of knowledge. Meanwhile they had no 
influence upon the schools, although they were thoroughly adapted to 
dispel the old prejudice against differential and integral calculus. 

I shall now emphasize only a very little of the further development of 
the nineteenth century. In the first place, I shall speak of a few advances 
which lie in the direction B: modern geometry, mathematical physics, 
along with theory of functions of a complex variable, according to Cauchy 
and Riemann. The leaders, in the first working over of these three 
great fileds, were the French. This is the place to say a word, also, about 
the style of mathematical presentation. In Euclid, one finds everything 
according to the scheme "hypothesis, conclusion, proof", to which is 
added, sometimes, the "discussion", i.e., the determination of the limits 
which the considerations are valid. The belief is widespread that 
mathematics always moves thus four steps at a time. But just in the 
period of which we are speaking, there arose, especially among the 
French, a new art form in mathematical presentation, which might be 
called artistically articulated deduction. The works of Monge or, to mention 
a more recent book, the Traite d y Analyse, by Picard, read just like a 
well written gripping novel. This is the style which fits the method of 
thought B, whereas the Euclidean presentation is related, in essence, to 
the method A. 

Of Germans who achieved distinction in these fields I should mention 
Jacobi (18041851), Riemann (18261866), and, coming to a somewhat 
later time, Clebsch (18331872), and the Norwegian Lie (1842-1899). 
These all belong essentially to the direction B, except that occasionally an 
algorithmic touch is noticeable with them. 

From the middle of the century on, the method of thought A comes 
again to the front with Weierstrass (1815 1 897) . His activity, as teacher 
in Berlin, began in 1856. I have already instanced Weierstrass function 
theory as an example of A. The more recent investigations concerning the 
axioms of geometry belong, likewise, to the type A. One is concerned 
here with studies entirely in the Euclidean direction, which approach it, 
also, in the manner of presentation. 

Brief Survey of the History of Mathematics. 85 

With this I bring our brief historical resume to an end. Many points 
of view which could only be alluded to here will be brought up later for 
more complete discussion. As a summary, we might say that, in the history 
of mathematics during the last centuries, both of our chief methods of investiga- 
tion were of importance] that each of them, and sometimes the two in suc- 
cession, have resulted in important advances of the science. It is certain 
that mathematics will be able to advance uniformally in all directions, 
only if neither of the two methods of investigation is neglected. May each 
mathematician work in the direction which appeals to him most strongly. 

Instruction in the secondary schools, however, as I have already 
indicated, has long been under the one-sided control of the Plan A. 
Any movement toward reform of mathematical teaching must, therefore, 
press for more emphasis upon direction B. In this connection I am 
thinking, above all, of an impregnation with the genetic method of 
teaching, of a stronger emphasis upon space perception, as such, and, 
particularly, of giving prominence to the notion of function, under fusion 
of space perception and number perception! It is my aim that these 
lectures shall serve this tendency, especially since these elementary 
mathematical books to which we are in the habit of going for advice, 
e g., those of Weber- Wells tein, Tropfke, M. Simon, represent the direc- 
tion A almost exclusively. I called your attention, in the introduction, 
to this one-sidedness. 

And now, gentlemen, enough of these diversions; let us pass to the 
next main subdivision of this course. 

Part II 


Let me commence by mentioning a few textbooks of algebra, in order 
to introduce you somewhat to a very extensive literature. I suggest, 
first, Serret's Cours d'algebre 1 which was much used in Germany, formerly, 
and had great merit. Now, however, we have two great widely used 
German textbooks: H. Weber's Lehrbuch der Algebra 2 and E. Netto's 
Vorlesungen uber Algebra*, each in two volumes; both treat with great 
fullness the most difficult parts of algebra and are well adapted for 
extensive special study; they seem to me to be too comprehensive for 
the average needs of prospective teachers and also too expensive. More 
fitting in the latter respect is the handy Vorlesungen uber Algebra* by 
G. Bauer, which hardly goes beyond what the teacher should master 5 . 
On the practical side, for the numerical solution of equations, this book 
is supplemented by the little book Praxis der Gleichungen by C. Runge 6 , 
which I can highly reccomend. 

Turning now to the narrower subject, let me remark that I cannot, 
in the limits of this course of lectures, give a systematic presentation of 
algebra] I can give, rather, only a one sided selection, and it will be most 
fitting if I emphasize those things which are, unfortunately, neglected 
elsewhere, and which are calculated nevertheless to throw light upon 
school instruction. All of my algebraic developments will group them- 
selves about one point, namely, about the application to the solution of 
equations of graphical and, generally speaking, of geometrically perceptual 
methods. This field alone is a very extensive and widely related chapter 
of algebra. Even from it, it is obviously possible to select only the most 

1 Third edition. Paris 1866 [sixth edition, 1910]. 

2 Second edition. Braunschweig 1898/99. [New revision by R. Fricke. Vol. I. 

3 Leipzig 1896/99. See also: Chrystal, Textbook in Algebra (2 volumes). 
Macmillan. Bocher, M., Introduction to Higher Algebra. Macmillan. 

[ 4 Second edition. Leipzig 1910.] 

5 See also: Netto, E., Elementare Algebra, akademische Vorlesungen fur 
Studierende der ersten Semester. [Second edition. Leipzig 1913, and H.Weber, 
Lehrbuch der Algebra. Small edition in one volume. Second printing. Braunschweig 
1921.] See also: Fine, H., College Algebra. Ginn. Hall und Knight, Higher 
Algebra. Macmilian. 

[ 6 Second edition. Leipzig 1921.] See also: v. Sanden, H., Practical Mathemat- 
ical Analysis. Button & Co. 

Equations with one Parameter. 7 

important and interesting things; in doing this we shall come into 
organic relation with the most widely differing fields, so that we shall 
be studying mathematics quite in the spirit of our system B. In the first 
place, we shall treat equations in real unknowns in order that we may 
follow, later, with the consideration of complex quantities. 

I. Real Equations with Real Unknowns 

1. Equations with one parameter 

We begin with a very simple case, which is susceptible of geometric 
treatment, namely with a real algebraic equation for the unknown x, 
in which a parameter A appears: 

/M) = 0. 

We shall obtain a geometric representation most simply if we replace A 
by a second variable y and think of 

f(*,y) = o 

as a curve in the xy plane (see Fig. 19). The points of intersection of 

this curve with the line y = A, parallel to the x-axis, give the real roots 

of the equation / (#, A) = . When we have 

drawn the curve approximately, as we can 

easily do if / is not too complicated, we can 

see at a glance by displacing the parallel 

as A varies, how the number of real roots 

changes. This plan is especially effective 

when / is linear in A, i.e. with equations 

of the form 

Fig. 19- 

<p(x) Ay(#) = 

If <p and y are rational, the curve y = y (x)/ip (x) will also be rational, and 
is easy to draw. In these cases one can often use this method to ad- 
vantage in calculating approximately the roots of equations. 
As an example consider the quadratic equation 

x 2 + ax A = 0. 

The curve y = x* + ax is a. parabola, and one can see at once for what 
values of A the equations has two, one, or no real roots according as the 
horizontal line cuts the pafabola in two, one, or no points (see Fig. 20). 
It seems to me that the presentation of such a simple and obvious con- 
struction would be very appropriate in the upper school classes. 
As a second example let us take the cubic equation 

x* + ax* + bx A = 0, 

which gives us the cubical parabola y = x 3 + ax 2 + bx, whose appear- 
ance is different according to the values of a , b . In Fig. 21 , it is assumed 

Algebra: Real Equations with Real Unknowns. 

that x 2 + ax + b = has two real roots. It is easy to see how the 
parallels group themselves into those which intersect the curve in one 

Fig. 20. 

Fig. 21. 

point and those which meet it in three ; there can be two limiting positions 
which yield double roots. 

2. Equations with two parameters 

When several parameters, let us say two, appear in an equation, 
more skill is required to handle the problem graphically, but the results 
are more extensive and interesting. We shall limit ourselves to the 
case where the two parameters h , p appear linearly, and we shall write t 
for the unknown in the equation. The problem is to determine the real 
roots of the equation 

(1) 9>W+*'Z0+fvW ='0, 
where <p, %, y> are polynomials in t. 

If x, y are ordinary rectangular point-coordinates, every straight 
line in the x y plane will be given by an equation of the form 

(2) y + ux + v = Q. 

We may call u, v the coordinates of the straight line. Then ( u) is the 
trigonometric tangent of the angle which the line makes with the 

tan <p u 
Fig. 22. 

and ( v) is the y-intercept 
(see Fig. 22). Let us think of points 
and lines as of equal importance; 
and let us give equal attention to 
point coordinates and line coordi- 
nates. This will be especially impor- 
tant later on. Then we may say that 
the equation y + MX + v = indicates 
the united position of the line (u , v) 
that the point lies on the given line, 

and of the point (x, y), i. e 

and the line goes through the given point. 

In order now to interpret the equation (1) geometrically, let us 
identify it with (2). This can be done in two essentially different ways 
which we shall consider, separably. 

Equations with two Parameters. 
A. Let us consider the equations 


v ' 


If t is variable, the equations (3 a) represent, a witt determined rational 
curve of the xy plane, which is called the normal curve of equation (1). 
Since every point on it corresponds to a definite value of t, a certain 
scale of values of t is defined upon it. By means of (3 a) we can calculate 
as many points as we please; and hence we can draw the normal curve, 
with its scale, as accurately as we please, say on millimeter paper. 
For every definite pair of values of A and p (3b) represents a straight 
line of the plane. From what has been said, it follow that (1) shows, 
that the point t of the normal curve lies upon this straight line. Thus 
we obtain all the real roots of (1) if we find all the real intersections of the 
normal curve with this line and read off their parameter values on the 
curve scale. The normal curve is determined, once for all, by the form 
of equation (1), regardless of the special values which the parameters 4, /a 
may have. For every equation with definite A , // there is, then, one 
straight line which represents it, in the manner described above, so that, 
in general, all the straight lines in the plane 
come into play, whereas before (pp. 8788) 
only horizontal lines were used. 

As an illustration, let us take the quad- 
ratic equation 

< 2 + J< + A* = 0. 

The normal curve here is given by the 

y = t 2 , x = t or y = x 2 , 
i.e., the normal curve is the parabola shown 
in Fig. 23, with the scale there indicated. 

We can at once read off the real roots of our equation as the inter- 
sections with the line y + A# + ^ = 0. In particular, the figure 
shows that the two roots of the equation t 2 t \ = lie between 
f and 2, and between and 1 , respectively. The essential advantage 
of this method, over that given on pp. 8788, is that we can now solve 
all quadratic equations with one and the same parabola, if we make use 
of all the straight lines in the plane. Thus, if we wish to solve, approxi- 
mately, a considerable number of equations, one can apply this method 
very effectively. 

In a similar way one can treat the totality of cubic equations, all of 
which can, by a linear transformation, be thrown into the reduced form 

t* + it + fi = 0. 

Fig. 23. 


Algebra: Real Equations with Real Unknowns. 

The normal curve here is the cubical parabola 

= t or 

y = 

sketched in Fig. 24. This method also seems to me to be usable in the 

schools. The pupils would certainly derive pleasure from drawing such 


B. The second method of interpreting (1) is got from the first by 

applying the principle of duality, i. e., by interchanging point and line 

coordinates. To that end, let us write 
the terms of (2) in reverse order: 

v + %u + y = 

and identify it, in this form, with (1) 
by setting 

/ W 

< /w v / 



Y I A; ii 

x A , y ft, . 

If t is variable, the equation (4 a) 
represents a family of straight lines 
which will envelope a definite curve, 
Fig. 24. the normal curve of (1), in the new 

interpretation. It is a rational class 

curve, since it is represented, in line coordinates, by rational functions of 
a parameter. Every tangent, and hence the corresponding point of tan- 
gency, is determined by a definite value of t, so that one gets again a 
scale on the normal curve. By drawing a sufficient number of tangents 
according to (4 a), we may draw both curve and scale with any desired 
degree of exactness. Each parameter-pair A , /u, determines, by virtue 
of (4b), a point in the xy plane, through which, by virtue of (1), the 
tangent t of the normal curve (4 a) must pass. We obtain, therefore, all 
the real roots of (\) by reading off the parameter-values t belonging to all 
the tangents to the normal curve which go through the point x = A , y p . 
As before, the normal curve is completely determined by the form of 
equation (1). Every equation of this form will be represented, for given 
values of the parameters i , // , by a certain point in the plane, or, if 
we wish, by its position with respect to the curve. 

Let us illustrate by means of the same examples as before. Corres- 
ponding to the quadratic equation 

P + It + /LI = 
the normal curve will be the envelope of the straight lines 

This envelope, again, is a parabola with its vertex at the origin. The 
graph, drawn on fine cross section paper exhibits immediately the real 

Equations with two Parameters. 


roots of t 2 + It + ft = as parameters t of the tangents drawn to the 
parabola from the point A,^ (see Fig. 25). 
For the cubic equation 

, / 3 + A* + ^-0 

the normal curve 

v = t*, u = t 

will be a curve of the third class with a cusp at the origin, shown in 
Fig. 26. 

We can present this method somewhat differently. If we examine 
the so-called trinomial equation 

t + lp + p^Q, 

we may represent the system of tangents to the normal curve by means 
of the parameter equation 

f(t) =^ + ^ + y =,:0. 

Fig. 25. 

The equation of the normal curve in point coordinates may be found, 
as usual, by eliminating t between the last equation and the equation 
obtained by differentiation with respect to t\ 

/' (t) = mt m ~ l + nxt n - l = 

for the normal curve, as the envelope of the system of straight lines, 
is the locus of the intersection of each of these lines with the neighboring 
line (for t and t + dt). If, instead of eliminating /, we express x and y 
as functions of t from these two equations, we find 

(5 a) 

y - 

which are the point equations of the normal curve: 

As normal curves for the quadratic and the cubic equations which 
were selected above as examples, one finds in this way, respectively, 

These are the curves which are sketched in Figs. 25 and 26. 


Algebra: Real Equations with Real Unknowns. 

Fig. 27. 

Let me emphasize the fact that this method is put to practical use 
by C. Runge, in his lectures and exercises, and that it has proved itself 
especially appropriate for the actual solution of equations. We might 
profitably use one or the other of these graphical methods in school 

If we now compare with each other the two methods which we have 
developed, we find that, for at least one definite and very important 
purpose, the second offers a distinct advantage, namely, when one seeks 
a visible representation of all the equations of a definite type which have 
a given number of real roots. Such totalities are represented, according 
to the first method, by systems of straight lines ; according to the second, 
however, by fields of points. But because of the peculiar nature of our 

geometric perception, or of our habit, the 
latter are essentially easier to grasp than are 
the former. 

I shall show at once, by means of the 
example of the quadratic equation, what can be 
done in this direction (see Fig. 27). From all 
points outside of the parabola two tangents 
can be drawn to the curve; from points 
within, none. Hence these two regions represent 
the manifolds of all equations with two roots and with no roots, respectively. 
For all points of the parabola itself there is only a single tangent, which 
can be counted twice. The normal curve itself is, then, in the general case, 

the locus of those points whose coordinates I , ft 
yield equations with two equal roots, so that 
we may call it the discriminant curve. 

In the case of the cubic equation, we see 
that from a point inside the angle of the 
1 normal curve one can draw three tangents 
to the curve. This is obvious for points on 
the median line, because of symmetry; and 
the number cannot change when the point 
varies, provided it does not cross the curve. 
If the point (x, y) moves to the curve, two 
of the tangents coincide; if it moves into 

the region outside the curve, both of these tangents become imaginary 
and there remains but one real tangent. Accordingly, the region inside 
the angle of the normal curve represents the totality of cubic equation with 
three different real roots ; the region outside, equations with only one real 
root', while to the points on the curve itself correspond the equations with 
one simple and one double real root. Finally, a triple tangent goes 
through the cusp, corresponding to the single equation / 3 = 0, with a 
single triple root. Figure 28 makes this obvious at a glance. 

Fig. 28. 

Equations with two Parameters. 


The pictures become much more interesting and show more, if, as 
is customary in algebra, we impose definite restrictions upon the roots, 
in particular, if we inquire about all the real roots lying within a given 
interval t l <it^t 2 . As you know, the general answer to this question 
is furnished by Sturm's theorem. We can, however, easily complete our 
drawing so that it will give a satisfying graphical solution of this general 
question also. For this purpose we simply add to the normal curve the 
tangents to it determined by the parameter values t lf t 2 and consider the 
division of the plane into fields which these tangents bring about. 

To carry through these considerations for the quadratic equation, 
we must determine the number of tangents which touch the parabolic arc 
between t and t 2 . Through every point of the triangle (see Fig. 29) 
bounded by the parabolic arc and these two 
tangents there are obviously two tangents. 
If the point crosses either of the tangents 
t lf t 2 , one of the tangents through it will 
touch the parabola beyond the arc (t, t 2 ), 
and so will be lost for our purpose. Tangents 
from points which lie within the two crescent 
shaped areas bounded by the parabola and 
the tangents t lt t 2 touch the parabola outside 
the arc (t t 2 ) ; and from points within the parabola there are no real 
tangents at all. The two parabolic arcs t ^ t and t^>t% are thus of 
no significance in effecting the desired subdivision of the plane. There 
remain, then, only those lines in the figure 
which are drawn full; these, together with 
the numbers assigned to them, give at a 
glance exact information as to the manifolds 
of quadratic equations which have 2, 1, or 
real roots between t and t 2 . 

We may proceed similarly with the cubic 
equation (see Fig. 30). Let us take, say, t > 0, 
/ 2 < . Again we draw the tangents with these 
parameter values and examine the subdivi- 
sions of the plane brought about by them 
and the arc of the normal curve which lies 

between ^ and t 2 . Through every point in the four-cornered region at 
the cusp there will be three real tangents which touch between t and t 2 . 
If point crosses either of the tangents ^ , t 2 , there is a loss of one tangent 
of this character. When it crosses the normal curve two are lost. From 
these considerations we obtain the picture, shown in Fig. 30, of the 
regions of the plane which correspond to equations with three, two, one, 
or no roots lying between t and t 2 . In order to see the great usefulness 
of the graphical method, one need only make a single attempt to picture 

Fig. 30. 

Q4 Algebra: Real Equations with Real Unknowns. 

abstractly this classification of cubic equations, without making any 
appeal whatever to space perception ; it will require a disproportionately 
great amount of time. And the proof, which here becomes evident by 
a glance at the picture, will not be at all easy. 

Now as to the relation of this geometric method to the well known 
algebraic criteria of Sturm, Cartesius, and Budan-Fourier I remark, 
merely, that the geometric method includes them all, for equations of 
the types which we have considered. You will find these relations 
carried out more fully in my article 1 "Geometrisches zur Abzdhlung der 
Wurzeln algebraischer Gleichungen" and in W. Dycks "Katalog mathe- 
matischer Modelle" 2 . I am glad to take this occasion to refer you to 
this catalog. It was published on the occasion of the exposition, in 
Munich, in 1893, by the German Mathematical Society, and remains 
today the best means of orientation in the field of mathematical models. 

3. Equations with three parameters A, fi, v 

Finally, I shall also show you that one can apply analogous considera- 
tions to equations with three parameters. We shall need to use space 
of three dimensions instead of the plane. It will suffice if I consider the 
special equation of four terms 

The method of procedure can be applied immediately to equations of 
other forms. 

In addition to this equation, we shall use the condition, from space 
geometry, that a point (x , y , z) and a plane with the plane coordinates 
(u,v,w) shall be "in united position", i.e., that the plane (u,v,w) 
shall contain the point (x,y t z). This condition is 

(2) z + u% + vy + w = 

(3) w + xu + yv + z = 0. 

We now identify this equation, written in the one form or the other, 
with (1) and we obtain, exactly as before, two mutually dual inter- 

Let us then set 
(2 a) z = P, oc = t m , y = t n . 

These equations determine a certain space curve, the normal curve of 
the four -term equation (1), together with a scale of the values t. Then we 

[* Reprinted in Klein, F., Gesammelte Mathematische Abhandlungen, vol. II, 
pp. 198-208.] 

2 A catalogue of mathematical and mathematical-physical models, apparatus, 
and instruments (Munich, 1892), also a supplement to this (Munich, 1893). 

Equations with three Parameters A, ,a, v. 


consider the plane which is determined by the coefficients A, fi, v , 

of (1): 

(2b) 71 = A, v = /LI, w v. 

Then equation (1) says that the real roots of the proposed equation are 
identical with the parameter values t of the real intersections of the normal 
curve (2 a) with the plane (2b). 

If we choose the method dual to the preceding, we must put 

(3a) w = t p , u = t m , v = f. 

These equations represent, for 
variable t, a simple infinity of 
planes, which we can look upon 
as the osculating planes of a 
definite space curve associated, 
as before, with a scale of para- 
meter values t. This will be a 
normal f< class curve" , being ex- 
pressed in plane coordinates, in 
distinction from the previous 
normal "order curve'', which 
was given in point coordinates. 
If we now consider, in conjunc- 
tion with the first curve, also 
the point 

(3b) x = l, y = t*, z = v, 

it follows that the real roots of (1) 
are identical with the parameter 
values t of those osculating planes 
of the normal class curve (3 a) 
which pass through the point (3 b). 

Let us next illustrate these two interpretations by concrete examples. 
We have, in our collection, models for both of them, which I shall now 
put before you. 

The first method was used by R. Mehmke, in Stuttgart, in the con- 
struction of an apparatus for the numerical solution of equations. His 
model is a brass frame (see Fig. 31) in which you will notice three 
vertical rods carrying scales, and into which one can fit curved templates, 
or stencils, of the normal curves of equations of degree three, four, or 
five, (after these have been reduced to four terms). Note, however, that 
while our exposition presupposed the ordinary rectangular coordinate 
system, Mehmke has so determined his coordinate system that the appro- 
priate plane coordinates, i.e., the coefficients u, v, w of the equation of 
the plane (2), are precisely the intercepts which this plane makes on the 

96 Algebra: Real Equations with Real Unknowns. 

scales of the three vertical rods and which one can read off there. In 
order, now, to make possible the fixation of a definite plane u = A, 
v = /LI , w = v, a peep-hole is provided on the w rod, which one sets 
at the reading v of that scale, while one joins by a stretched string the 
readings, of the scales on the u and v rods, respectively. The rays joining 
the peep-hole with this string make our plane, and by looking through 
the peep-hole one can observe directly the intersections of this plane with 
the normal curve as the apparent intersections of the string with the template. 
Their parameter values, the desired roots of the equation, are read at the 
same time on the scale of the normal curve, which is affixed to the template. 
The practical usableness of this apparatus depends, of course, upon 
the carefulness of its mechanical construction, but the limited power 
of accommodation of the human eye would, at best, make it very 

For the second method, a model was prepared by Hartenstein in con- 
nection with his state examination. It has to do with the so-called 
reduced form of the equation of degree four, that is, 

t* + U^ + iJit + v=Q, 

to which every biquadratic equation can be reduced. I shall present 
this method in a form somewhat different from the one I used for the 
two-parameter equation (p. 91). In the present case we have to consider 
a simple infinity of planes whose plane coordinates are given in (3 a) 
and whose point equations would be written as follows: 

(4) /(*) =P + xt* + yt + * = 0. 

The envelope of these planes is the system of the straight lines in which 
each plane / (t) = meets the neighboring plane / (t + dt) = 0, i.e., 
the developable surface whose equation is obtained by eliminating t between 
f (t) = and /' (t) = . But in order to obtain the normal curve we must 
seek the osculating configuration of the system of planes, i.e., the locus 
of the points of intersection of three successive planes. This locus is, as 
you know, the cuspidal edge of that developable surface and its coordinates 
are found, as functions of t , from the three equations f (t} = , /' (t) = , 
f" (t) = 0. In our case these three equations are: 

t* + xt 2 + yt + z --= Q 
4t* + x-2t + y = 
12* a + *-2 =0, 

and one finds from them: 

(5) x = -6t*, y = 8t*, * = -3. 

These expressions represent the point equation of the normal class curve 
of (4) whose plane equation, by (3 a), may be written in the form 

(6) w = t*, u = t*, v = t. 

Equations with Three Parameters A, fi, v. 


Both forms are of degree four in t. Hence the normal curve is both of 
order four and of class four. 

In order to study it more in detail, let us consider a few simple 
surfaces which pass through it. In the first place, the expressions (5) 
satisfy identically in t the equation 

Hence our normal curve lies upon a parabolic cylinder of order two 
whose generators are parallel to the y-axis. Likewise, we have the relation 

~8~ ~*~ "27 ~~ ' 

so that this cubic cylinder, whose generators are parallel to the z-axis, 
also goes through our normal curve. Moreover, the normal curve is the 
finite intersection of these two cy- 
linders. With these facts in mind, 
one can form an approximate 
picture of the course of the nor- 
mal curve. Is is a skew curve, 
symmetric to the x z plane, having 
a cusp at the origin (see Fig. 32). 
Again the quadric surface 

x-z _ 3y 2 _ 

6 64 

goes through our normal curve; 
for, by (5), this equation is also Fig. 32. 

satisfied identically in /. From 

it, and the equation of the cubic cylinder, we find another linear 
combination which represents an especially important surface of the 
third degree passing through the normal curve: 

^? Z_ 2 _ 5 3 _ _ o 
6 16 216 ~~ 

Let us now consider the developable surface whose cuspidal edge is 
the normal curve, and which we can define as the totality of the tangents 
to the normal curve. The tangent at the point t to any space curve 

is given by the equations 

* = v(*) + Q<P' W > y = v W + e W , * = * W + ez'W 

in which Q is a parameter. For the direction cosines of the tangents 
to the curve are to each other as the derivatives of the coordinates 
with respect to / . lit is thought of as variable, we have in these equations, 
with two parameters t, Q , the representation of the developable surface. 

Klein, Elementary Mathematics. 7 

Qg Algebra: Real Equations with Real Unknowns. 

All this follows from well known theorems of space geometry. For our 
curve (5) we get, in particular, the following equations for the developable 
surface. If we call the coordinates of its points (X , Y , Z) to distinguish 
them from the coordinates of the curve, the equations of the develop- 
able are 


Now this surface is the basis of the Hartenstein model, its straight lines 
being represented by stretched threads (see Fig. 33)- 

The parameter representation offers the best starting point for the 
discussion and the actual construction of the surface. Indeed, it is only 
from force of habit that we inquire about the equation of the surface 
itself. We can obtain it by eliminating Q and t from (7). I shall give 
you the simplest procedure for this without giving the details of the 
inner meaning of the several steps. From (7) we form the combination 

X 7 V 2 Jf 3 
A lf __ J*_ ___ ^L 

6 16 216 

both of which vanish on the curve itself (for Q = 0). If we equate 
these to zero, we obtain two of the surfaces mentioned above which 
pass through the curve. Eliminating the product Qt from these equations, 
we find the equation of the developable surface 

( 7J _X*\* ^( x ' z y2 x3 Y*-n 

r + i2J ~~ 27 r 6~ ~ 16 ~ iiej " - 

The surface is thus of order six; but it is composed of the plane at in- 
finity and a surface of order five. 

As to the meaning of this formula, I make the following remark for 
those who are acquainted with the subject. The expressions in the two 
parentheses are the invariants of the biquadratic equation 

t* + Xt*+ Yt + Z = 0, 

with which we started. These play an important role in the theory of 
elliptic functions and they are designated there, in general, by g 2 an( i 3- 
The left side of the equation of our surface, A = gij 27 gL is, as you 
know, the discriminant of the biquadratic equation, which indicates, by 
its vanishing, the presence of a repeated root. Our developable surface 
is therefore the discriminant surface of the biquadratic equation, i.e., the 
totality of the points for which it has a double root. 

After these theoretical explanations, the construction of a thread 
model for our surface offers no essential difficulty. By means of the 
parameter equations (7) we may determine, say, the points in which 

Equations with Three Parameters 


those tangents which we wish to represent intersect certain fixed planes. 
We then stretch threads between these planes, which are made out of 
wood or cardboard. But it requires long trial and great skill to make 
the model really beautiful and usable, and to bring out the entire inter- 
esting course of the surface and of its cuspidal edge, as in the model 
before us. The sketch on page 99 (see Fig. 33) shows the surface with 
its straight lines; AOB is the cuspidal edge [see the figure p. 97 1 ]. 

You notice on the model a double curve (COD) 
along which two sheets of the surface intersect. This 
curve is simply the following parabola of the 
X Z plane : 

Only one half (CO) of this parabola, namely that 
for X < 0, appears, however, as the intersection 
of real sheets, while the other half lies isolated in 
space. This phenomenon is by no means sur- 
prising to those who are accustomed to illustrate 
the theory of algebraic surfaces by concrete geo- 
metric representations. It is a common thing, 
there, for real branches of double curves to appear 
both as intersections of real sheets and also in 
part isolated. In the latter 
case we regard them as real 
intersections of imaginary sheets 
of the surface. The correspond- 
ing phenomenon in the plane 
is more generally known. In 
that case, in addition to the 
ordinary double points of al- 
gebraic curves, which appear 
as intersections of real bran- 
ches of the curve, there are also the apparently isolated double points, 
which may be regarded as the intersections of imaginary branches. 
Let us now make clear in detail, what this surface with its cuspidal 
edge, the normal curve, can do for us. We think of the normal curve 
with its associated scale, or, better, we affix to each tangent its para- 
meter value /, which also belongs to the point of tangency. //, now, 
someone gives us a biquadratic equation with definite coefficients (x,y,z), 
we need only to pass through the corresponding point (x,y,z)the osculating 
plane to the normal curve, or, what would be the same thing, the tangent 

1 The Hartenstein string model was put upon the market by the firm of M. Schil- 
ling in Leipzig. A dissertation by R. Hartenstein entitled: Die Discriminanten- 
fldche der Gleichung vierten Grades goes with the model Leipzig, Schilling, 1909. 

Fig. 33. 

1QO Algebra: Real Equations with Real Unknowns. 

plane to the discriminant surface, to obtain the real roots as the parameter 
values of the points of contact with the curve, or the parameter values of 
the corresponding tangents, as the case may be. Since the osculating plane 
cuts the curve where it touches it, every point of contact of an osculating 
plane with the curve is projected from the point (x, y , z) as an apparent 
point of inflexion on the curve, and conversely. Consequently, the real 
roots of the biquadratic equation are, finally, the parameter values t of 
the apparent inflexion points of the normal curve, viewed from the point 
(x, y , z) in space. 

Now it is, of course, quite difficult for the unpractised eye to deter- 
mine with certainty from the model either the planes of osculation or 
the apparent inflexions of the curve. But the model exhibits with 
immediate clearness the next important thing, the classification of all 
biquadratic equations according to the number of their real roots. Let us 
see, by ^n abstract examination of equations, just what cases one might 
expect . If <x , /? , y , d are the four roots of the real biquadratic equation (4) , 
then & + /? + y + ^=0, because of the vanishing of the coefficient 
of / 3 . So far as the reality of the roots is concerned, the following 
principal cases are possible: 
I. Four real roots. 
II. Two real, and two conjugate complex roots. 

III. No real, and two pairs of conjugate complex roots. 

If, now, two equations of the type I are proposed, with roots a, , /?, y , <5 
and <*', /?', y', 6', respectively, then one certainly could transform a, /J, 
y, d continuously into <x', /?', y', <5', respectively, through systems of 
values whose sum is always zero, At the same time, the one equation 
would transform continuously into the other, through equations always 
of the same type, i.e., all equations of type I make up a connected 
continuum, and the same is true for the other two types. Our model 
must therefore exhibit space partitioned into three connected parts such that 
the points in each part correspond to equations of one type. 

Let us now consider the transition cases between these three sorts. 
Type I goes over into II through equations which have two different real 
roots and one double (i. e. two coincident) real root, which we shall indicate 
symbolically by 2 + (2) ; similarly we have between II and III the 
transition case of one real double root and two complex roots, which may 
be indicated by (2). To both of these sorts there must correspond, in our 
model, regions of the discriminant surface, which, indeed, pictures all 
equations with coincident roots. Considerations similar to those above 
would show that to each type there must correspond a connected region of 
this surface. Now, again, these two groups, 2 + (2) and (2), go over 
into each other by means of cases with two real double roots, symbolically : 
(2) + (2); the points for which two pairs of roots move thus into co- 
incidence must belong simultaneously to two sheets of the discriminant 

The Fundamental Theorem of Algebra. -101 

surface, that is, to the non isolated branch of the double curve. Accordingly, 
the discriminant surface falls into two parts, separated by a branch of the 
double curve] one of these parts, 2 + (2), separates the space regions I 
and II, the other, (2), the space regions II and III. In order to see, now, 
how the normal curve lies, we notice that, because of its property as 
a cuspidal edge, three tangent planes must merge into one (the osculating 
plane) at each point on it, so that we have the case of a triple and a 
simple real root: 1 + (3) . This can happen only when one of the simple 
roots becomes equal to the double root. Consequently, the cuspidal edge 
must lie entirely on the first part, 2 + (2), of the surface. In the cusp of 
the cuspidal edge (x = y = z = 0) we have a quadruple real root, which 
can arise from the case (2) + (2) through the coincidence of the two 
double roots. In fact, the cusp, , of the cuspidal edge lies also on the 
double curve. Finally, as to the isolated branch of the double curve, it lies 
entirely in the space region III and is characterized by the fact that 
on it the two pairs of conjugate complex roots merge into one complex 
double root. Both double roots are, of course, conjugate to each other. 
You can recognize on our model all of the possible cases enumerated 
above. In the sketch (Fig. 33> P- 99), the interior of the surface to 
the right of the double curve is region I, to the left, region III; the 
exterior is region II. You will be able easily to become fully oriented 
by means of the following tabulation, which exhibits the number and 
the multiplicity of the real roots which correspond to the points of the 
several space, surface, and curvilinear regions. In this scheme, the digits 
not in parentheses denote the number of simple real roots, the others, 
as before, denote the multiplicity of repeated roots: 

I. II. III. 

Region : 
Discrim. surface: 
Normal curve: 
Double curve 
Cusp : 

4 2 

2+ (2) 
1 -H3) 


(2) + (2) 

(2imag. double roots). 


II. Equations in the field of complex quantities 

We shall now remove the restriction to real quantities and shall 
operate in the field of complex quantities. Of course, we shall endeavor 
again only to emphasize those things which are susceptible of geo- 
metric representation to an extent greater than one finds elsewhere. 
Let us begin at once with the most important theorem of algebra. 

A. The fundamental theorem of algebra 
This is, as you know, the theorem that every algebraic equation of 
degree n in the field of complex numbers has, in general, n roots, or, more 

102 Algebra: Equations in the Field of Complex Quantities. 

accurately, that every polynomial f (z) , of degree n, can be separated into n 
linear factors. 

All proofs of this theorem make fundamental use of the geometric 
interpretation of the complex quantity x + iy in the x y plane. I shall 
give you the train of thought of Gauss' first proof (1799), which can be 
presented quite graphically. To be sure, the original exposition of Gauss 
was somewhat different from mine. 

Given the polynomial 

./(*) = z n + a lZ n - l + ... + a n , 
we may write 

f(x + iy) = u (x, y) + i v (x, y) , 

where u , v are real polynomials in the two real variables x . y . The 
leading thought of Gauss' proof lies now in considering the two curves 

u (x , x} = and v (x , y) = 

in the x y plane, and in showing that they must have one point, at least, 
in common. For this point one would then have / (x + iy) = 0, that 
is, the existence of a first "root" of the equation f = would be proved. 
For this purpose, it turns out to be sufficient, to investigate the be- 
haviour of both curves at infinity, i.e., at a distance from the origin 
which is arbitrarily great. 

If r , the absolute value of z, is very large, we may neglect the lower 
powers of z in / (z) , in comparison with z n . If we introduce polar co- 
ordinates r, <p into the x y plane, i. e., if we set 

z = r (cos (jp + i sin q>) , 
we have, by De Moivre's formula 

z n = r n (cosnq) + isinwep). 

This expression is approached asymptotically by / (z) , as z increases 
in absolute value. It follows at once that u and v approach, respectively, 
asymptotically the functions 

r n cosn(p, r n smn<p. 

Consequently the ultimate course of the curves u = 0, v = 0, at in- 
finity, respectively, will be given approximately by the equations 

cos n <p = , sinn<p = 0. 

Now the curve sin n q> = consists of the n straight lines which 
go through the origin and make with the #-axis the angles 0, nfn, 
2 7i In , . . . , (n 1) n/n , whereas cos n<p = Q consists of the n rays through 
the origin which bisect these angles (Fig. 34 is drawn for n = 3). In 
the central part of the figure, the true curves u = 0, v = can, of 
course, be essentially different from these straight lines; but they must 
approach the straight lines asymptotically as the lines recede from the 

The Fundamental Theorem of Algebra. 


origin. We can indicate their course schematically by retaining the straight 
lines outside of a large circle and replacing them by anything we please, 
inside the circle (see Fig. 35). But no matter what the behavior of the 

Fig. 34. 

curves may be inside the circle, it 
is certain that, if one makes the 
circle about the origin sufficiently 
large, the branches u , v , outside 
the circle, must alternate, from 
which it is graphically clear that 
these branches must cross one another 
inside the circle. In fact, we can 
give a rigorous 1 proof of this 
assertion, and this is the sub- 
stance of Gauss' proof if we use 
the continuity properties of the 
curves. The preceding argument, 
however, gives the essentials of 
the train of thought. If one such 
root has been found, we can divide 
out a linear factor, and we can then 


Fig. 36. 

1 It should be said here that Gauss does not dispense entirely with geometric 
considerations. The arithmetization of the proof which he contemplated in his 
dissertation was first given by A. Ostrowski (Gottinger Nachrichten, 1920, or 
vol. VIII of the materials for a scientific biography of Gauss, 1920). It is of 
historical interest that the first proof of the fundamental theorem was by D'Alem- 
bert. To be sure, there was an error in his proof, to which Gauss called attention. 
D'Alembert, namely, failed to distinguish between the upper limit of a function 
and its maximum, and he made use of the assumption, which in general is false, 
that a function of a complex variable actually attains its upper limit when this 
limit exists. N 

1Q4 Algebra: Equations in the Field of Complex Quantities. 

repeat the reasoning for the other polynomial factor of degree (n 1). 
Continuing in this way, we may finally break up f (z) into n linear factors, 
i. e., we may prove the existence of n zeros. 

This method of reasoning will be much clearer if you carry through 
the construction for special cases. A simple example would be 

/ ( z ) = z * - \ = . 

In this case we obviously have 

u = ? 3 cos399 1 , v = r 3 sin3<p, 

so that v = consists simply of three straight lines, while u = has 
three hyperbola-like branches. Figure 36 shows the three intersections 
of the two curves, which give the three roots of our equation. I re- 
commend strongly that you work through other and more complicated 

These brief remarks about the fundamental theorem will suffice 
here, since I am not giving a course of lectures on algebra. Let me 
close by pointing out that the significance of the admission of complex 
numbers into algebra lies in the fact that it permits a general statement 
of the fundamental theorem. With the restriction to real quantities 
one can only say that the equation of degree n has n roots, or fewer, 
or perhaps none at all. 

B. Equations with a complex parameter 
The rest of the time which I have set aside for algebra I shall devote 
to the discussion, by graphical methods, of all the roots (including the 
complex ones] of complex equations, as was done earlier for the real roots 
of real equations. We shall limit ourselves, however, to equations with 
one complex parameter and we shall assume, furthermore, that this 
occurs only linearly. The study of a simple con formal representation will 
then give us all that is required. 

Let z = x + iy be the unknown, and w = u + iv the parameter. 
Then the type of the equation to be considered has the form 

(1) <p(z) w-ip(z) = 

where (p , y , are polynomials in z . Let n be the highest power of z that 
occurs. According to the fundamental theorem, this equation has for 
each definite value of w exactly n roots z which, in general, are different. 
Conversely, however, it follows from (1) that 

i.e., w is a single-valued rational function of z, and it is said to be of 
degree n. If we should use, as geometric equivalent of equation (1), 

Equations with a Complex Parameter. 

simply the conformal representation which this function sets up between 
the 2-plane and the z^-plane, the many-valuedness of z as function of w 
would be visually disturbing. We may help ourselves here, as is always 
the case in function theory, by thinking of the w-plane as consisting 
of n sheets, one over another, which are united in an appropriate manner, 
by means of branch cuts, into an n leaved Riemann surface. Such surfaces 
are familiar to you all from the theory of algebraic functions. Then our 
junction establishes, between the points of the n-leaved Riemanris surface 
in the w-plane and the points of the simple z-plane, a one-to-one relation 
which is, in general, conformal. 

Before we begin a detailed study of this representation, it will be 
helpful if we set up certain conventions which will do away with the 
exceptional role played by infinite values of w and z, a role not justified 
by the nature of the case, and which will enable us to state theorems 
in general form. Inasmuch as these conventions are not so widely 
employed as they should be, you will permit me to say a word or two 
more about them than I otherwise should. We cannot be satisfied here 
when one speaks merely symbolically of an infinitely distant point of the 
complex plane, since such a conception gives no adequate concrete 
image, so that one must have recourse to special considerations or stipula- 
tions, in order to find out what corresponds, for an infinitely distant 
point, to a definite property of a finite point. But we can secure all that 
is desired, if we replace the Gaussian 
plane, as picture of the complex num- 
bers, once for all, by the Riemannian 
sphere. For this purpose, we think 
simply of a sphere of diameter one, 
tangent to the % y plane, its south 
pole S being at the origin, and 
from its north pole N we project 
the plane stereographically upon 
the sphere (see Fig. 37). To every 
point Q = (x, y) of the plane there 
corresponds uniquely the second Fig. 37. 

intersection P of the ray NQ 

with the sphere; and, conversely, to every point P of the sphere, 
with the exception of N itself, there corresponds a unique point Q 
with definite coordinate (x, y). Hence we can consider P as representing 
the number x + iy. Now if P approaches the north pole N, in any 
manner, Q moves to infinity ; conversely, if Q recedes to infinity in any 
manner, the corresponding point P approaches the single definite 
point N. It seems natural, then, to look upon this point N, which does 
not correspond to any finite complex number, as the unique representative 
of all infinitely large x + iy, i.e., as the concrete picture of the infinitely 

406 Algebra: Equations in the Field of Complex Quantities. 

distant point of the plane, which is otherwise introduced only symbolically, 
and to affix to it outright the mark <x>. In this way we bring about, in 
the geometric picture, complete equality between all finite points and the 
infinitely distant point. 

In order to return now to the geometric interpretation of the 
algebraic relation (1), we shall replace the w plane also by a w-sphere. 
Then our function will be represented by a mapping of the z-sphere 
upon the w-sphere, and, just as in the case of the mapping of the 
two planes, this is also conformal, since the stereographic mapping 
of the plane upon the sphere is, according to a well known theorem, 
conformal. To a single position on the w- sphere, there will then 
correspond, in general, n different positions on the 2-sphere. In order 
to get a one-to-one relation we imagine, again, n sheets on the 
z0-sphere, lying one above another, and united, in appropriate manner, 
by means of branch cuts, so as to form an n-leaved Riemann surface 
over the w-sphere. This picture presents no greater difficulty that that 
of the Riemann surface over the plane. Thus, finally, the algebraic 
equation (1) is interpreted as a one-to-one relation, conformal in general, 
between the Riemann surface over the w-sphere and the simple surface 
of the z-sphere. This interpretation obviously takes into account, also, 
infinite values of z and w which may correspond to each other or to 
finite values. 

In order to make the greatest possible use this geometric device, 
we must take a corresponding step in algebra, one which shall do away 
with the exceptional role which infinity plays in the formulas, and this 
step is the introduction of homogeneous coordinates. We set, namely, 

and consider z^ , z 2 as two independent complex variables, both of which 
remain finite, and which cannot both vanish simultaneously. Each definite 
value of z will then be given by infinitely many systems of values 
(cz lt cz^, where c is an arbitrary constant factor. We shall look upon 
all such systems of values (cz lt cz 2 ) which differ only by such a factor, 
as the same "position" in the field of the two homogeneous variables. 
Conversely, for every such position there will be a definite value of z, 
with one exception : to the position (^ arbitrary, z 2 = 0) there will 
correspond no finite z; but if one approaches it from other positions, 
the corresponding z becomes infinite. This one position is thus to be 
looked upon as the arithmetic equivalent of the one infinitely distant point 
of the z-plane or, as the case may be, of the z-sphere, and as carrying the 
mark z = oo. 

In the same way, of course, we put also w = w l /w 2 . We shall now set 
up the "homogeneous" equation between the "homogeneous" variables z l9 

Equations with a Complex Parameter. 107 

z 2 and w lf w 2 , which corresponds to equation (2). Multiplying by z% in 
order to clear of fractions, we may write the equation in the form 

In this equation, q> (z l , 2 2 ) and y (^ , 2 2 ) are rational integral functions 
of z and, z z , since 9? (2) and^ (z) contain at most the nth power of z = z l /z 2 - 
Moreover they are homogeneous polynomials (forms) of dimension n. 
For each term z 1 of (p (z) or y (z) is transformed into the term 
Z2(z- L lz 2 ) i = * ""**! of dimension w, by clearing of fractions. 

We come now to the detailed study of the functional dependence which 
our equation (1) or, as the case may be, (3) establishes between z and w. 
We shall apply consistently our two new aids, mapping upon the complex 
sphere and homogeneous coordinates. We shall have solved this problem 
when we can form a complete picture of the conformal relation between 
the ^-sphere and the Riemann surface over the z^-sphere. 

First of all we must inquire as to the nature and the position of the 
branch points of the Riemann surface. I remind you here that a /J-fold 
branch point is one in which ^ + 1 leaves are connected. Since w is 
a single-valued function of z, we know the branch points when we know 
the points of the z sphere which correspond to them, which I am in the 
habit of calling the critical or noteworthy points of the z-sphere. To 
each of these there corresponds a certain multiplicity equal to that of 
the corresponding branch point. I shall now give, without detailed 
proof, the theorems which make possible the determination of these 
points. I assume that the rather simple functiontheoretic facts which 
enter into consideration here are in general familiar to you, though they 
may not be in the homogeneous form which I prefer to use. I shall 
illustrate in concrete graphical form the abstract considerations which 
I shall present to you, in this connection, by a series of examples. 

A little calculation is necessary in order to obtain the analogue, in 
homogeneous coordinates, of the differential coefficient dw/dz. Differ- 
entiating equation (3) and omitting the bars over q> and y, we obtain 

_ y>d<p 

We have also 

dq) = 

dip = 


Algebra :" Equations in the Field of Complex Quantities. 

On the other hand, from Euler's theorem for homogeneous functions of 
degree n, we have 

9?i #1 -f- % ' ^2 = w <p 

= n 

consequently the numerator on the right side of (3') may be written 
in the form 


d<p, dip 
<P> V 


<p l dz l -f 

This expression, by the multiplication theorem for determinants, becomes 

Thus (3') goes over into the equation 

This constitutes the basal formula of the homogeneous theory of our 
equation, and the functional determinant (p l y 2 <P% V ; i of the forms cp , ip 
appears as a crucial expression for all that follows. Except for it and 
for the factor 4/(^ V 2 ) one has on the right the differential of z = z^z^, 
on the left that of w = wjw^. Since for finite z and w the critical 
points are given by dw/dz = 0, as is well known, the following theorem 
appears plausible, but I shall here omit the proof. Each /i-fold zero of 
the functional determinant is a critical point of multiplicity //, i.e., 
there corresponds to it a /Li-fold branch point of the Riemann surface over 
the w-sphere. The chief advantage of this rule, as compared with those 
which are otherwise given, lies in the fact that it contains in one statement 
both finite and infinite values of z and w. It enables us also to make 
a precise statement concerning the number of remarkable points. The four 
derivatives, namely, are forms of dimension n \ , and the functional 
determinant is therefore a form of dimension 2 n 2 . Such a polynomial 
always has 2 n 2 zeros, if one takes into account their multiplicity. 
Thus, if HI, & 2 , . . ., oc v are the remarkable points of the z-sphere (i.e., if 

their respective multiplicities, then their sum is 

it _i- II J- . . . _L- II O*f "> 

jt/j [^ A 2 1^ T^ r"V ~* * 

By virtue of the conformal mapping, to these points there correspond 
the v branch points 

a l , a 2 , . . . , a v 

on the Riemann surface over the w-sphere, which must necessarily lie 
separated on the surface, and about which fa + 1 , fa + 1 , . . . , /* v + \ 
leaves, respectively, must be cyclically connected. It should be noted, 

Equations with a Complex Parameter. 109 

however, that different ones of these branch points may lie over the 
same position on the w sphere, since w = <p (z)/y(z) for z = <x lt <x 2 . , . . , <x v 
may give the same value for w more than once. Over such a point, 
there would be two or more separate series of leaves, each series being 
in itself connected. Every such position on the w sphere is called a 
branch position; we shall denote them, in order, by A , B , C , . . . . It 
should be noted that their number can be smaller than v. 

The statements thus far made furnish only a hazy picture of the 
Riemann surface. We shall now build it up so that it can be more readily 
visualized. For this purpose, let us draw on the w sphere through the branch 
positions A , B , C , . . . an arbitrary closed curve ( without double points 
and of the simplest possible form (see Fig. 38), and distinguish the two 
spherical caps thus formed as the upper cap and 
the lower cap. In all of the examples which 
we shall discuss later the points A , B , C , . . . 
will all be real and we shall then naturally 
select as the curve ( the meridian great circle of 
real numbers, so that each of our two partial 
regions will be a hemisphere. 

Returning to the general case we see that 
each pair of leaves of the Riemann surface 
which are connected, intersect along a branch rig. 33. 

cut which joins two branch points. As you 

know, the Riemann surface remains unchanged in essence if we move 
these cuts, leaving the end points fixed, that is, if we think of the 
same leaves as being connected along other curves, provided these 
join the same branch points. It is in just this variability that the 
great generality and also the great difficulty of the idea of the Rie- 
mann surface lies. In order to give the surface a definite form, which 
shall be susceptible of concrete visualization, we move all the branch cuts 
so that all of them lie over the curve & , which passes through all the branch 
points. It may be that several branch cuts lie over the same part of 
(, and none at all over other parts. 

Now let us cut this entire complex of leaves, i.e., each individual leaf, 
along the curve (. Since we had already moved all the branch cuts into 
position over (, the incision just made passes along all of them, so that 
our Riemann surface separates into 2n "half -leaves" entirely free from 
branches, n of them over each of the two spherical caps. If we think of 
the half -leaves corresponding to the upper cap as being shaded, and those 
corresponding to the lower as not shaded, we can distinguish briefly, 
n shaded and n unshaded half -leaves. We can now describe the original 
Riemann surface as follows. On it each shaded half-leaf meets only un- 
shaded half -leaves, those with which it is connected along segments of the 
curve ( lying over A B , B C , . . . ; and, similarly, each unshaded half-leaf 

-J-JQ Algebra: Equations in the Field of Complex Quantities. 

is connected along such segments of ( only to shaded half -leaves. However, 
more than two half -leaves may meet only at a branch point] and in fact 
around any [t-fold branch point, fi + 1 shaded half-leaves would alternate 
with ft + 1 unshades ones. 

Since the mapping by means of our function w (z) of the z sphere 
upon the Riemann surface over the w sphere is a one-to-one correspon- 
dence, we can immediately transfer to the z sphere the above conditions 
of connectivity. Because of continuity, the 2 n half-leaves of the Rie- 
mann surface must correspond to 2n connected z regions, which we 
may call the shaded and the unshaded half-regions. These will be 
separated from one another by the n images of each of the segments 
AB, BC , . . . oi the curve ( which the w-valued function z (w) represents 
upon the z sphere. Each shaded half-region meets only shaded half-regions 
along these image-curves, and each unshaded half-region meets only shaded 
ones. It is only in a ft-fold critical point that more than two half-regions 
can meet. At such a point (JL + 1 shaded and // + \ unshaded half -regions 
come together. 

This division of the z sphere into partial regions will help us to follow 
in detail the course of the function z (w) for a few simple characteristic 
examples. I shall begin with the simplest one possible. 

1. The "pure" equation 
We shall call the well known equation 

(1 ) z n = w 

a pure equation. Its solution is given formally by introducing the 

w r~ 
radical z = \w. This gives us no information, however, regarding the 

functional relation between z and w . We shall proceed according to the 
general plan by introducing the homogeneous variables 

-^i i I ' 
w* ~ z\* 

and we shall consider the functional determinant of the numerator and 
denominator of the right side 

This expression obviously has the (n 1) fold zeros z l = and z 2 = 0, 
or (in non-homogeneous form) 2 = and z = oo. These are the only 
critical points and they are of total multiplicity 2n 2. By our 
general theorem, therefore, the only branch points of the Riemann surface 
over the w sphere are at the positions w = and w = <x>. By the equation 
w z n these correspond to the two points z = and z = oo . Each 
of these two points has the multiplicity n \ , so that n leaves are 

The "Pure" Equation. 



cyclically connected at each of them. Let us now mark on the w sphere 
the meridian of real numbers as the curve and let us cut all the leaves 
of the Riemann surface along this meridian, after having appropriately 
displaced all of the branch cuts. Of the 2n hemispheres into which the 
surface separates we think of those over the rear half 
of the w sphere, that is, those which correspond to w 
values with positive imaginary parts, as shaded. Upon 
the meridian itself, we shall distinguish between the 
half meridian of positive real numbers (drawn full in 
Fig. 39) and that of the negative real numbers (dotted). 
Now we must examine the mappings of this 
meridian ( curve upon the z sphere, where they bring 
about the characteristic division into half-regions. 
Upon the positive half meridian w = r, where r ranges 
through positive real values from tooo; for these values we have 
by a well known formula of complex numbers, 

n )' 

z = 

= Vr cos- 


+ isin- 

where & = 0, 1, ..., 1. 

For the different values of k , this expression gives those n half-meridians 
of the z sphere which make with the half-meridian of positive real numbers 
the angles 0, 2 n\n, 4^jn t . . ., 2(n l}n\n. Thus these curves corres- 


pond to the full drawn half of (. On the negative half -meridian of the 
w sphere we must set w = r = r e ijl , where again ^ r ^ oo. This 


^ - 


, . 

+ ism 

(2k + 



, where = 0, 1, . . .,n 1. 

Corresponding to this we have, on the z sphere, those n half-meridians 
which have the "longitude" n/n, $n/n, . . . ,2(n \)n/n, which thus bisect 
the angles between the others. Accordingly, the z sphere is divided into 
2n congruent sectors reaching from the north pole to the south pole, similar 

112 Algebra: Equations in the Field of Complex Quantities. 

to the natural divisions of an orange. This division is exactly in accord 
with the general theory. In particular, it is only at the remarkable 
points, the two poles, that more than two half -regions meet. At each 
of these points 2n half -regions meet, corresponding to the multipli- 
city n \ . 

As for the shading of the regions, we need to fix it for one region only. 
The remainder are then alternately shaded and unshaded. Now note 
that when we look at the shaded half of the w sphere (the rear) from the 
point w = 0, the full drawn part of the boundary lies to the left, the 
dotted part to the right. Since we are concerned with a conformal map- 
ping in which angles are not reversed, each shaded portion of the z sphere, 
looked at from the correponding point z , must have the same property 
as to position, that is, it must have a full drawn boundary to the left, and 
a dotted one to the right. With this we control completely the division 
of the z sphere into regions. Moreover, one notices a characteristic 
difference in the distribution of the regions upon two z hemispheres, 
according as n is even or odd, as can be clearly seen in Figs. 40 and 41 
on p. Ill for the first cases n = 3, n = 4. Let me emphasize how 
necessary it was to go over to the complex sphere in order to get a full 
understanding of the situation. In the complex z plane, one would 
have a division into angular sectors by straight lines radiating from 
z = 0, and it would not be at all so obvious that z = oo and w oo 
have equal significance with z = and w = 0, as critical point 
and branch point, respectively. 

This furnished us with the essentials for exact knowledge of the 
functional relation between z and w. We need now study only the 
conformal mapping of each of the 2 n spherical sectors upon one or the 
other of the two w hemispheres. But I shall not go into the details here. 
This case, as one of the simplest and most obvious illustrations, will 
be familiar ground to any one who has had to do with conformal re- 
presentation. We shall see later (see p. 131) how to deduce from this 
methods for the numerical calculation of z. 

Let us, however, settle here the important question as to the mutual 
relation among the various congruent regions of the z sphere. Speaking more 
exactly, w = z n takes on the same value at a point in each one of the n 
shaded regions. Can the corresponding values of z be expressed in terms 
of one another? We notice, in fact, that for z' = z (where e is any 
one of the nth roots of unity) z' n = z n , that is w = z n takes the same 
value at all the n positions 


= g'' . z = e n -z (v = 0, 1, 2, . . . , n 1). 

These n values of z' must therefore be distributed so that just one of 
them lies in each of the n shaded regions of the z sphere, if z is taken 

The "Pure" Equation. 

in one of the shaded regions and each of them must traverse one of 
these regions as z traverses its region. The same thing is true of the 
unshaded regions. Each of the substitutions (2) is represented geo- 
metrically by a rotation of the z sphere through an angle v 2 n\n about 
the vertical axis 0, oo, since, as is well known, multiplication in the 
complex plane by e 2vijT/n denotes a rotation through that angle about 
the origin. Thus corresponding points of our spherical regions, as well 
as the regions themselves, go over into one another by means of these n rota- 
tions about the vertical axis. 

If, then, we had determined at the start only one shaded partial 
region of the sphere, this remark would have furnished all the similar 
partial regions. In this we have made use only of the property of the 
substitutions (2) that they transform equation (\) into itself (i.e., z n = w 
into z tn = w) and that their number is equal to the degree. In the examples 
that follow, we shall always be able to give such linear substitutions 
at the outset, and by means of them to simplily the determination of 
the division into subregions. 

By using the present example I should like to illustrate an important 
general notion, namely, the notion of irreducibility for equations which 
contain a parameter w rationally. We have already discussed irreduci- 
bility of equations with rational numerical coefficients in connection with 
the construction of the regular heptagon (p. 51 et seq.). An equation 
f (z t w) = (e.g., our equation z n w = 0), where f (z t w) is a poly- 
nomial in z , whose coefficients are rational functions of w , is called reducible 
with respect to the parameter w , when f can be split into the product of 
two polynomials of the same sort, in each of which z really appears 

f (z,w) =f l (z,w) f 2 (z,w)', 

otherwise the equation is called irreducible with respect to w. The entire 
generalization, in comparison with the earlier conception, lies in the 
fact that the tf domain of rationality" in which we operate and in which 
the coefficients of the admissible polynomials are to lie, consists of the 
totality of rational functions of the parameter w instead of the totality of 
rational numbers, in other words, that we pass from a numbertheoretic 
to a functiontheoretic conception. 

If we illustrate this, for each equation / (z, w} = 0, by means of its 
Riemann surface, we can set up a simple criterion for reducibility in this 
new sense. If the equation, namely, is reducible, every system of the 
values z, w which satisfies it satisfies either f (z, w} = or / 2 (z, w) = 0; 
now the solutions of / t = and / 2 = are represented by means of 
their Riemann surfaces, which have nothing to do with each other, 
and, in particular, are not connected. Thus, the Riemann surface which 
belongs to a reducible equation f (z, w) = must break down into at least 
two separates pieces. 

Algebra: Equations in the Field of Complex Quantities. 

According to this, we can now assert that the equation z n w = 
is certainly irreducible in the function theoretic sense. For, on its Riemann 
surface, which we known exactly, all the n leaves are cyclically connected 
at each of its branch points. Moreover, the entire surface is mapped upon 
the unpartitioned z sphere. Hence such a breaking down cannot occur. 
In connection with this, we can answer one of the popular problems of 
mathematics which we touched earlier (p. 51), namely, that of the possibility 
of dividing an arbitrary angle <p into n equal parts, in particular, for n = 3, 
the possibility of trisecting an angle. The problem is to give an exact 
construction with ruler and compasses for dividing into three equal parts 
any angle (p whatever. (It is easy, of course, to give a construction for 
a series of special values of <p). I shall give you the train of thought 
for the proof of the impossibility of trisecting an angle in the sense just 

mentioned, and I shall ask you to recall, in 
this connection, the proof of the impossibility 
of constructing the regular heptagon with 
ruler and compasses (see p. 51 et seq.). Just 
as at that time, we shall reduce the problem 
to that of the solution of an irreducible cubic 
equation, and we shall then show that this 
equation cannot be solved by a series of 
Fig. 42. square roots; except that, now, the equation 

will contain a parameter (the angle (p) , whereas, 

before, the coefficients were integers. Accordingly, functiontheoretic 
irreducibility must replace numbertheoretic irreducibility. 

In order to set up the equation of the problem let us think of the 
angle (p as laid off from the positive real half -axis in the w plane (see Fig. 42) . 
Then its free arm will cut the unit circle in the point 

w = e i( v = cos9? + isincp. 

Our problem consists in finding, independently of special values of the 
parameter <p, a construction, involving a finite number of applications 
of the ruler and compasses, which shall give the point of intersection 
with the unit circle of the arm of the angle 90/3 , i. e., the point 

z = e 3 = cos Y + isin Y 

This value of z satisfies the equation: 

(3) z* = cosy + isiiKp , 

and the analytic equivalent of our geometric problem consists in solving 
this equation (see p. 51) by means of a finite number of square roots, 
one over another, of rational functions of sin (p and cos 9? , since these 
quantities are the coordinates of the point w with which we start the 

The Dihedral Equation. 

We must show, first, that the equation (3) is irreducible in the function 
theoretic sense. To be sure, this equation does not have just the form 
we assumed while explaining the notion, since, instead of the a complex 
parameter w that enters rationally, we have now two functions cos 
and sin of a real parameter <p, both of which appear rationally. As a 
natural extension here of our notion, we shall call the polynomial 
z 3 (cos <p + i sin q>) reducible if it can be split into polynomials whose 
coefficients are likewise rational finctions of co .; (p and sin (p ; and we 
can, as before, assign a criterion for this. If we let (p assume all real 
values in (3), w = e itp = cos 9? + i sin (p will describe the unit circle of 
the w plane, to which the equation of the w sphere corresponds by stereo- 
graphic projection. The curve which lies over this, on the Riemann 
surface of the equation 3 = w, and which describes, in one stroke, 
all three leaves, is mapped by equation (3) uniquely upon the unit 
circle of the z sphere. Hence it can be regarded, in a sense as its "one 
dimensional Riemann image". In the same way, we can obviously 
assign such a Riemann image to every equation of the form / (z , cos q> , 
sin <p) = by taking as many copies of the unit circle with arc length (p 
as the equation has roots, and joining them according to the connectivity 
of the roots. It follows, just as before, that the equation (3) can be reducible 
only when its one-dimensional Riemann image breaks down into separate 
parts, and this is obviously not the case. This proves the function theoretic 
irreducibility of our equation (3). 

Now, however, the former proof of the theorem, that a cubic equation 
with rational numerical coefficients is reducible if it can be solved by 
a series of square roots, can be applied literally to the present case of 
the function-theoretically irreducible equation (3) (see p. 51 et seq.). 
We need only to replace "rational numbers" there by "rational functions 
of cos <p and sin <p" . This proves our assertion that the trisection of an 
arbitrary angle cannot be accomplished by a finite number of applications 
of a ruler and compasses. Hence the endeavors of angle -trisection 
zealots must always be fruitless! 

I pass on now to the treatment of a somewhat more complicated 

2. The dihedral equation 
The equation 

is called the dihedral equation, for reasons that will appear later. 
Clearing of fractions, we see that its degree is 2n. Introducing homo- 
geneous variables we get 

116 Algebra: Equations in the Field of Complex Quantities. 

in which, in fact, forms of dimension 2n appear in numerator and 
denominator. The functional determinant of these forms is 

It has an (n l)-fold zero at 2 X = and at z a = 0; the other 2n zeros 
are given by 

4-zl = or: 
If in addition to the n-th root of unity 


E = e n 

which we have already used, we introduce also the primitive w-th root 
of -1: 


the last 2n zeros are given by the equations 

^ = e" and - 1 - = e- e v , (v = 0, 1 , . . . , n 1) . 

^2 Z 2, 

Since the values of z = ^/^ corresponding to them all have the absolute 
value one, they all lie therefore on the equator of the z sphere (corres- 
ponding to the unit circle of the z plane), at equal angular spacings of n\n . 
We have therefore as critical points on the z sphere: 

(a) the south pole z = and the north pole z = oo , each of multiplicity 

(b) the 2n equatorial points z = e r , f' e v , each of multiplicity one. 
The sum of all the multiplicities is 2 (n 1) + 2n \ = 4n 2, 
as is demanded by the general theorem on p. 108 for the degree 2n. 
By virtue of equation (1) there will correspond to the remarkable points 
z = o, z = oo of the z sphere, the position w = oo on the w sphere. 
Moreover, to all the points z = r , corresponds the position w = +1 ; 
and, to all the points z = e t? the position w = \ . There are, accord- 
ingly, only three branch points <x>, +1, \ on the w sphere. These 
will lie as follows: 

w = oo two branch points of multiplicity n \ ; 
w = +1 branch points of multiplicity 1; 
w = i w branch points of multiplicity 1. 

TA0 2w Ztfflfltfs o/ /As Riemann surface group themselves therefore over 
the point w = oo in two separate series, each of n cyclically connected 
leaves', over w = +1 and w = 1 iw n series, each of two leaves. The 
disposition of the leaves will become clear when we study the corres- 
ponding subdivision of the z sphere into half-regions. 

The Dihedral Equation. 


To this end it will be well, as we remarked above, to know the linear 
substitutions which transform equation (1) into itself. As in the case of 
the pure equation, it is unchanged by the n substitutions 

( 23 -) 

n \), where e = e 

since for these z' n = z n . Likewise, however, it is unchanged by the n 
additional substitutions 


z' = ~(v = Q,\, ...n-\). 

since these only change z n into \/z n . 

We have therefore 2n linear substitutions of equation (1) into itself, 
exactly as many as its degree indicates. Thus, if we know for a given 
value W Q of w one root Z Q of the equation, we know immediately 2 n roots 


^- Sphere: 

v Z Q and e v /z (v = , 1 , 2 , . . . , n 1 ) , in general all different, for which w 
has the same value W Q , i. e., we know all the roots of the equation when 
we have obtained the n-th root of unity e . 

Let us now proceed to examine the subdivision of the z sphere corres- 
ponding to cuts along the real meridian of the Riemann surface over the 
w sphere. In this, as in the previous example,we distinguish on the real 
meridian of the w sphere the three segments made by the branch points 
that from +1 to oo (drawn full), that from oo to 1 (short dotted), 
and that from 1 to +1 (long dotted) (see Fig. 43). To each of these 
three segments there correspond on the z sphere 2n different curvilinear 
segments which can be derived from any one of them by means of the 2 n 
linear substitutions (2). It will always suffice, therefore, to find one of 
them. Moreover all these segments must connect the critical points 
z = o, oo, e v , e' e v , which we therefore mark on the z sphere. Just as 
in the previous case, their form is of a somewhat different type according 
as n is even or odd. It will suffice if we exhibit a definite case, say for 
n = 6. Fig. 43 shows the front half of the z sphere in orthogonal pro- 
jection. One sees, on the equator, from left to right with spacings of 

H8 Algebra: Equations in the Field of Complex Quantities. 

60, 3 = 1 , 4 , 5 , e 6 = 1 ; and lying midway between the others, e 7 - e 3 , 
E' 4 = i , and e' 5 . 

A/oze> w shall see that the quadrant +1 < 2 < oo of the meridian of 
real z corresponds to the part of the real w meridian +1 <w<oo (full 
drawn). In fact, if we put z = r and let r range through real values 
from 1 to oo, then w = i (z n +\jz n } = \ (r n + \jr n ) will vary also through 
real values that are always increasing, from 1 to oo . We obtain n other 
full drawn curves on the z sphere, from this one, by means of the n linear 
substitutions (2 a). But, as we saw in the previous example, these 
substitutions mean rotations of the sphere about the vertical axis (0 , oo) 
through the angles 2njn, 4n/n, . . . , 2 (n 1) n\n. We get in this way 
the n quarter-meridians from the north pole oo to the points r on the 
equator. We get an additional full drawn curve if we apply 
the substitution z' = \\z, which transforms the meridian quadrant 
from +1 to oo into the lower real meridianquadrant from 
+ 1 to 0. If we subject this quadrant to the n rotations (2 a), 
the composition of which with z' = \jz gives the n substitutions 
(2b), we obtain, in addition, the n meridian quadrants which join the 
south pole with the equatorial points e v . We have now in fact the 2n 
full drawn curves which correspond to the full drawn w meridian qua- 
drant. In particular, for n = 6, they make up the three entire meridians 
into which the real meridian is transformed by rotations of 0, 60, 120. 

It is now also obvious that the totality of the values z = e' r, 
where r again ranges through real values from -\-i to oo, corresponds 
to the dotted part of the real w meridian; for the equation (1) yields then: 

_ 1 / 'n w I 1 \ __ 1 /n M 

2 \ ' / n /yii i 2 \ ' /y n I * 

and this expression actually decreases through real values from 1 to 
oo. But z = e' r represents the meridian quadrant from oc to the 
equatorial point e v . If we now apply to it the substitutions (2 a), (2b), 
we find, as before, that to the dotted part of the real w meridian there corres- 
pond all the meridian quadrants joining the poles to the equatorial points 
e e v , which thus bisect the angles between the meridian quadrants which 
we obtained before. In particular, for n = 6, they make up the three 
entire meridians into which the real meridian is transformed by 
rotations of 30, 90, 150. 

There remain to be found the 2n curvilinear segments which corres- 
pond to the long-dotted half-meridian 1<10<+1. I shall prove 
that they are the segments of the equator of the z sphere determined by 
the points e v and e' e v . In fact, the equator represents the points of 
absolute value one and is given therefore by z = e i<p where (p is real 
and ranges from to 2n. Hence we have 


The Dihedral Equation. \\g 

This expression is always real, and its absolute value is not greater than 1 , 
In fact, it assumes once every value between + 1 and 1 as <p varies 
from one multiple of n\n to the next one, i.e., when z traverses one of 
the segments of which we are speaking. 

The curves determined in this manner divide the z sphere into 2 2 n 
triangular half-regions which are bounded by one curve of each of the three 
sorts, and each half-region corresponds to a half leaf of the Riemann surface. 
Several regions can meet only at the critical points, and then in accord- 
ance with the table of multiplicities (p. 116), namely, 2n at the north 
pole, and at the south pole, and 2 2 at each of the points e 1 ' and e' e v . 
In order to determine which of these regions are to be shaded, we notice 
that when w traverses, in order/ the full-drwan, the long-dotted, and 
the short dotted parts of the real w meridian, the rear half of the w sphere 
lies at its left. Since the mapping is conformal with preservation of 
angles, we should shade those half-regions whose boundaries follow 
one another in this same sense, and we should leave the others unshaded. 

We have now obtained a complete geometric picture of the mutual 
dependence between z and w which is set up by our equation. We might 
follow it out in greater detail by Examining more closely the conformal 
mapping of the single triangular regions upon the w hemisphere, but we 
shall forego this. / shall describe only, and briefly, the case n 6,to which 
I have already given special attention. The z sphere is then divided into 
twelve shaded and twelve unshaded triangles of which six of each sort 
are visible in Fig. 44. Six of each sort meet at each pole, and two of 
each sort at each of twelve equidistant points of the equator. Each 
triangle is mapped conformally upon a w half -leaf of the same sort. Of 
the half-leaves of the Riemann surface, six of each sort are connected 
at the branch position oo , and two of each sort at each of the branch 
positions ^ 1 > corresponding to the grouping of the half-regions on 
the z sphere. 

We may obtain a convenient picture of the division of the z sphere, 
and one which is especially valuable because of its analogy with pictures 
soon to come, as follows. If we join the n equidistant points on the 
equator (e. g., the e") with one another in order by straight lines, 
and also join each of them to the two poles, one obtains a double pyramid, 
with 2n faces, inscribed in the sphere (in Fig. 44, twelve faces). If we 
now project, from the center, the subdivision of the z sphere upon this 
double pyramid, every pyramid face is divided into a shaded and an 
unshaded half by the altitude of that face dropped from the pole. If 
we represent the division of the z sphere, and consequently our function, 
by means of this double pyramid, the latter will render a service quite 
analogous to that which we shall get in the coming examples from the 
regular polyhedra. We obtain a complete analogy if we think of the double 
pyramid as collapsed into its base, and consider the double regular n-gon 


Algebra: Equations in the Field of Complex Quantities. 

(hexagon) which results whose two faces (upper and lower) are divided 
each into 2 n triangles by the straight lines which join the center with 
the vertices and the middle points of the sides (see Fig. 45). / have 
been in the habit of calling this figure a dihedron and of classing it with 
the five regular polyhedra which have been studied since Plato's time. 
It fulfills, in fact, all the conditions by means of which a regular poly- 
hedron is usually defined, since its faces (the two faces of the w-gon) 
are congruent regular polygons, and since it has congruent edges (the 
sides of the n-gon) and congruent vertices (the vertices of the n-gon). 
The only difference is that it does not bound a proper solid body but 
encloses the volume zero. Thus the theorem of Plato, that there are 

Fig. 44. 

Fig. 45. 

only five regular solids, is correct only when one includes in the definition 
the requirement of a proper solid, which is always tacitly assumed in 
the proof. 

// we start with the dihedron, we obtain our subdivision of the z sphere 
by projecting upon that sphere not only its vertices but also the centers of 
its edges and its faces, the projecting rays for the latter being perpendi- 
cular to the plane of the dihedron. Thus the dihedron can also be looked 
upon as representing the functional relation which our equation sets up 
between w and z. Hence the brief name which we have already used, 
dihedral equation, is appropriate. 

In addition, we shall now consider those equations which, as already 
intimated, are closely related to the platonic regular solids. 

3. The tetrahedral, the octahedral, and the icosahedral equations. 
We shall see that the last two could, with equal right, be called the 
hexahedral and the dodecahedral equations, so that all five regular 
bodies will have been covered. We shall follow here a route that is 
the reverse of the one we followed in the preceding example. Starting 
from the regular body, we shall first deduce a division of the sphere into 
regions, and we shall then set up the appropriate algebraic equation, for 
which that figure is the proper geometric interpretation. I shall have to 
confine myself frequently to suggestions, however, and I therefore refer 
you at once to my book: Vorlesungen ilber das Ikosaeder und die Auf- 

The Tetrahedral, the Octahedral, and the Icosahedral Equations. \2\ 

losung der Gleichungen vom funften Grade 1 , in which you will find a 
systematic presentation of the entire extensive theory with its numerous 
relations to allied fields. 

Moreover, I shall give a parallel treatment of all three cases and 
I shall begin by deducing the subdivision of the sphere for the tetrahedron. 

\. The tetrahedron (see Fig. 46). We divide each of the four equi- 
lateral face-triangles of the tetrahdron, by means of the three altitudes, 
into six partial triangles. 
These are congruent in 
two groups of three each, 
while any two non- 
congruent ones are sym- 
metric. We obtain thus 
a division of the entire 
surface of the tetrahedron 
into twenty -four triangles, 
which fall into two groups, 
each containing twelve 

Face Triangle (actual size). 


congruent triangles, while Fig. 46. 

any triangle of one group 

is symmetric to every triangle of the other group. We shall shade the 

triangles of one group. Among the vertices of these twenty-four 

triangles we can distinguish three sorts, such that each triangle has one 

vertex of each sort: 

a) the four vertices of the initial tetrahedron, at each of which three 
shaded and three unshaded triangles meet', 

b) the four centers of gravity of the faces, which determine again 
another regular tetrahedron (the co-tetrahedron) ; at each of these, three 
triangles of each kind meet', 

c) the six middle points of the edges, which determine a regular octa- 
hedron', at each of these, two triangles of each kind meet. 

If from the center of gravity of the tetrahedron we project this subdivision 
into triangles upon the circumscribed sphere, the latter will be subdivided 
into 2 12 triangles, which are bounded by arcs of great circles and are 
mutually congruent or symmetric. About each vertex of the sort a), b), c), 
there will be respectively 6, 6, 4 equal angles, and since the sum of the 
angles about a point on a sphere is 2^, each of the spherical triangles 
will have an angle rc/3 at a vertex of the sort a or b and an angle n/2 at a 
vertex of the sort c. 

It is a characteristic property of this division of the sphere that it, 
as well as the tetrahedron itself, is transformed into itself by a number 

1 Leipzig 1884; referred to hereafter as "Ikosaeder". Translation into English 
by G. C. Morrice: Lectures on the Icosahedron by Klein. Revised Edition, 1911, 
Kegan Paul & Co. 


Algebra: Equations in the Field of Complex Quantities. 

of rotations of the sphere about its center. This will be clear to you in 
detail if you examine a model of the tetrahedron with its divisions, 
like the one in our collection. For the lecture, it will suffice if I indicate 
the number of possible rotations (whereby the position of rest is included 
as the identical rotation. If we select a definite vertex of the original 
tetrahedron, we can, by means of a rotation, transform it into every 
vertex of the tetrahedron (including itself), which gives four possibilities. 
If we keep this vertex fixed, however, in any one of these four positions, 
we can still transform the tetrahedron. This gives altogether 4 3 = 12 
rotations which transform the tetrahedron, or the corresponding tri- 
angular division of the circumscribed sphere, into itself. By means of 
these rotations we can transform a preassigned shaded (or unshaded) 
triangle into every other shaded (or unshaded) triangle, and the particular 
rotation is determined when that second triangle is chosen. These 
twelve rotations form obviously what one calls a group G 12 of twelve 
operations, i.e., if we performs two of them in succession, the result 
is one of the twelve rotations. 

If we think of this sphere as the z sphere, each of these twelve 
rotations will be represented by a linear transformations of z, and the 

twelve linear transformations which 
arise in this manner will transform 
into itself the equation which cor- 
responds to the tetrahedron. For pur- 
poses of comparison, I remark that 
one can interpret the 2 n linear sub- 
stitutions of the dihedral equation as 
the totality of the rotations of the 
dihedron into itself. 

2. We shall now treat the octa- 
hedron similarly (see Fig. 47) and 
we may be somewhat briefer. We 
divide each of the faces, just as 
before, into six partial triangles and 
obtain a division of the entire surface of the octahedron into twenty-four 
congruent shaded triangles, and twenty-four unshaded triangles which are 
congruent among themselves but symmetric to the other twenty-four. We 
can again distinguish three sorts of vertices: 

a) the six vertices of the octahedron, at each of which four triangles 
of each kind meet\ 

b) the eight centers of gravity of the faces, which form the vertices of 
a cube\ at each of these, three triangles of each kind meet; 

c) the twelve mid-points of the edges, at each of which two triangles 
of each kind meet. 

The Tetrahedral, the Octahedral, and the Icosahedral Equations. -123 

If we pass now to the circumscribed sphere, by means of central pro- 
jection, we obtain a division into 2 24 spherical triangles which are 
either congruent or symmetric, and each of which has an angle rc/4 at 
the vertex a , nfy at the vertex b , and n/2 at the vertex c . Since the 
vertices b form a cube, it is easy to see that one would have obtained the 
same division on the sphere if one had started with a cube and had projected 
its vertices, and the centers of its faces and edges, upon the sphere. In other 
words, we do not need to give special attention to the cube. 

Just as in the previous case, it is easy to see that the octahedron, 
as well as this division of 
the sphere t is transformed 
into itself by twenty-four 
rotations which form a group 
G 24 ; again each rotation is 
determined in that it trans- 
forms a preassigned shaded 
triangle into another definite 
shaded triangle. 

3. We come finally to 
the icosahedron (see Fig. 48) . 
Here, also, we start with 
the same subdivision of 
each of the twenty-four 
triangular faces and obtain 
altogether sixty shaded and 
sixty unshaded partial tri- 
angles. The three sorts of pig. 48. 
vertices are: 

a) the twelve vertices of the icosahedron, at each of which five triangles 
of each kind meet] 

b) the twenty centers of gravity of the faces, which are the vertices of a 
regular dodecahedron', at each of them three triangles of each kind meet] 

c) the thirty mid-points of the edges, at each of which two triangles of 
each sort meet. 

When this is carried over to the sphere each spherical triangle has 
at the vertices a t b t c the angles rc/5, nr/3, rc/2, respectively. From the 
property of the vertices b one can conclude, as before, that the same 
division of the sphere would have resulted if one had considered the dodeca- 

Finally, the icosahedron, as well as the corresponding division of the 
sphere, is transformed into itself by a group G 60 of sixty rotations of the 
sphere about its center. These rotations, as well as those for the octa- 
hedron, will become clear to you upon examination of a model. 

124 Algebra: Equations in the Field of Complex Quantities. 

Let me make a list of the angles of the spherical triangles which have 
appeared in the three cases which we have considered, to which I shall 
add the dihedron also; they are 

Dihedron : 7i/2 , n\2 , n\n ; 

Tetrahedron: aft, jr/3 , rc/2; 

Octahedron: n/4, n/3, n\2\ 

Icosahedron: jr/5, ft/3, n/2. 

As a variation of a joke of Kummer's I might suggest that the 
student of natural science would at once conclude from this, that 
there were additional subdivisions of the sphere, having analogous 
properties, and with angles such as Ji/6, nft, n\2\ n\7 , nft , 71/2. The 
mathematician, to be sure, does not risk making such inferences by 
analogy, and his cautiousness justifies itself here, for the series of possible 
spherical subdivisions of this sort ends, in fact, with our list. Of course 
this is connected with the fact that there are no more regular polyhedrons. 
We can see the ultimate reason in a property of whole numbers, which 
does not admit a reduction to simpler reasons. It appears, namely, 
that the angles of each of our triangles must be aliquot parts of n t 
say n/m, yi/n, n\r , such that the denominators satisfy the inequality 

\\m + \\n + \\r > 1 . 

This inequality has the property of existing only for the integral solutions 
given above. Moreover, we can understand it readily, since it only 
expresses the fact that the sum of the angles of a spherical triangle 
exceeds n. 

I should like to mention that, as some of you doubtless know, an 
appropriate generalization of the theory does carry one byeond these 
apparently too narrow bounds: The theory of automorphic functions in- 
volves subdividing the sphere into infinitely many triangles whose angle 
sum is less than or equal to n. 

4. Continuation: Setting up the Normal Equation. 
We come now to the second part of our problem, to set up that 
equation of the form 

(1) V (z)-u,v(,) = 0, or = , 

which belongs to a definite one of our three spherical subdivisions, that is, 
which maps the two hemispheres of the w sphere upon the 2-12, or 
the 2 24, or the 2 60 partial triangles of the z sphere. To each value 
of w there must correspond then, in general, 12, 24, 60 values, respectively, 
of z, each one in a partial triangle of the right kind. Hence the desired 
equation must have the degree 12, 24, 60 in the three cases respectively, 
for which we shall write N in general. Now each partial region touches 

Continuation: Setting up the Normal Equation. 



three critical points; hence there must be, in every case, three branch 
positions on the w sphere. We assign these, as is customary, to w = 0, 
1 , oo i and we choose again the meridian of real numbers as the section 
curve ( through these three points, whose three segments shall correspond 
to the boundaries of the z triangles. 

We shall assume (see Fig. 49) that in each of the three cases the 
centers of gravity of the faces (vertices b in the former notation) correspond 
to the point w = , the mid-point of the edges (vertices c) to the point w = \ , 
and the vertices of the polyhedron (vertices a) to the point w = <x>. The sides 
of the triangles will then correspond to the three segments of the w meri- 
dian in the manner indicated by the mapping, and the shaded triangles 
will correspond to the rear w hemisphere, the unshaded to the front w 
hemisphere. By virtue of these correspon- 
dences, the equation (1) is to effect a unique 
mapping of the z sphere upon an JV-leaved 
Riemann surface over the w sphere with 
branch points at , 1 , o . 

We might deduce, a priori, a proof for the 
existence of this equation by means of general 
functiontheoretic theorems. However, I prefer 
not to presuppose the knowledge which this 
method would require, but to construct the 
various equations empirically. This method 
will give us perhaps a more vivid perception 
of the individual cases. 

Let us think of equation (1) written in 
homogeneous variables 



Fig. 49- 

where <& N , 1 F N are homogeneous polynomials of dimension N in z lt z z 
(N = 12, 24, or 60). In this form of the equation, the positions w^ = 0, 
w 2 = (i.e., w = 0, oo) on the w sphere seem to be favored more than 
the third branch position w = 1 (in homogeneous form, w^ w 2 = 0). 
Since, however, the three branch positions are, for our purpose, of equal 
importance, it is expedient to consider also the following form of the 
equation : 

where X N = <&N Vy denotes also a form of dimension N. Both forms 
are embraced in the continued proportion 

(2) v>i : (w l w 2 ) :w 2 = $ N (z l , z 2 ) : X N (z l , * a ) : N (z l ,z 2 ). 

This furnishes us with a completely homogeneous form of equation (1) 
which gives the same consideration to all the branch points. 

126 Algebra: Equations in the Field of Complex Quantities. 

Our problem now is to set up the forms <&N , XN , Y N . For this purpose, 
we shall bring them into relation to our subdivision of the z sphere. 
From equation (2) we see that the form 0y (z lt z 2 ) = for w l = 0, i. e., 
that w = corresponds to the N zeros of &N on the z sphere. On the other 
hand, the centers of gravity of the faces of the polyhedron (vertices b in the 
subdivision), of which there are JV/3 in every case, must, according to 
our assumptions, correspond to the branch position w = 0. But every 
one of these centers must be a triple root of our equation, since in each 
of them there meet three shaded and three unshaded triangles of the 
z sphere. Thus these points, each with multiplicity three, supply all the 
positions which correspond to w = 0, and consequently all the zeros of 
&y. Hence <P N has only triple zeros and must, therefore, be the third 
power of a form <pn (z l , z 2 ) of degree Af/3 : 

In the same way, it follows that the zeros of XN = correspond to 
the position w = \ (i. e., w l w 2 = 0), and that these are identical with 
the N/2 midpoints, each counted twice, of the edges of the polyhedron 
(vertices c of our subdivision). Consequently XN must be the square 
of a form of dimension N/2: 

Finally the zeros of *P N are to correspond to the point 10 = 00, so that 
they must be identical with the vertices of the polyhedron (vertices a 
of the subdivision); but at these vertices 3, 4, or 5 triangles meet, in 
the several cases, so that we get 

YN = bp N!v (*i , * 2 )] v . where v = 3, 4 or 5 . 
Our equation (2) must then necessarily have the form 

(3) MI : K - w a ) w* = v(*i' Z 2? :*(*i. *a) a : V(*i> *a)"> 

where the degrees and powers of (p, %, y, and the values of the degree N 
of the equation are exhibited in the following table: 

Tetrahedron: <pl , jfa , yl ; N = 12. 
Octahedron: 9$ , % 2 12 , yg ; N = 24. 
Icosahedron : <p!! , xlo > v4 ; N = 60. 

I shall now show briefly that the dihedral equation which we discussed, 
fits also into the scheme (3). We need only to recall that in that case 
we chose 1 , +1, oo as the branch positions on the w sphere instead 
of 0, +1, oc which we selected later. We shall, then, obtain actual 
analogy with (3) only if we throw the dihedral equation into the form 

(u>i + w 2 ) :(w l w 2 ):w 2 =&:X: W. 

Continuation: Setting up the Normal Equation. 
Now from the dihedral equation (p. 115) which we used: 


we get by simple reduction 

( Wl + w 2 ) : K- ^ 2 ) : w 2 = (z\ 

2*1*3) : 

Thus we can, in fact, add to the above table: 

Dihedron : <pl, %*, yl\ N = 2n. 

The critical points together with their multiplicities which can at 
once be read off from this form of the equation are in full agreement 
with those which we found above (see p. 116). 

We come now to the actual setting up of the forms <p, %, y in the 
three new cases. I shall give details here only for the octahedron, for which 
the relations turn out to be the simplest. 
But even here I shall, at times, give only 
suggestions or results, in order to remain 
within the confines of a brief survey. For 
those who desire more, there is easily 
accessible the detailed exposition in my 
book on the icosahedron. For the sake of 
simplicity we think of the octahedron as 
so inscribed in the z sphere that the six 
vertices fall on (see Fig. 50) : 

z = 0, oo, + 1, +i, 1, - i. 

It will then be a simple matter to give the twenty-jour linear substitutions 
of z which represent the rotations of the octahedron, i.e., which permute 
these six points. We begin with the four rotations in which the vertices 
and oo remain fixed 

(4a) z' = i k -z, (6 = 0,1,2,3). 

Then we can interchange the points 0, oo by means of the substitution 
z' = \\z (i. e., a rotation through 180 about the horizontal axis (+1 , 1) 
which transforms every point of the octahedron into another one. If 
we now apply the four rotations (4 a), we get four new substitutions: 


I K 

(k = 0, 1, 2, 3) 

In the same way, we now throw in succession the four remaining vertices 
z = \ , i t 1 , i to oo by means of the substitutions 

z = 


z \ 

Z I' 


which obviously permute the six vertices of the octahedron, and again 

128 Algebra: Equations in the Field of Complex Quantities. 

apply, each time, the four rotations (4 a). Thus we get 4-4 = 16 ad- 
ditional substitutions for the octahedron 



z i ' z + i ' 

We have therefore found the desired twenty-four substitutions, and 
we can easily show, by calculation, that they really permute the six vertices 
of the octahedron and that they form a group G 24 , i. e., that the successive 
application of any two of them gives again one of the substitutions in (4). 

I shall now construct the form ^ 6 which vanishes in each of the 
vertices of the octahedron. The point z = gives the factor z lf the 
point z = oo the factor z 2 ; the form z\ z\ has a simple zero at each 
of the points 1 , i, so that we obtain finally 

(5a) ' y> 6 = z l z 2 (z\ z . 

It is more difficult to construct the forms <p s and # 12 which have 
as zeros the centers of gravity of the faces and the midpoints of the 
edges. Without deducing them, I may state that they are 1 

7>8 ^H 

It goes without saying that there is an undetermined constant 
multiplier in each of these three forms. If g? 8 , y e> # 12 stand for the 
normal forms (5), we must insert, in the octahedral equation (3), two 
undetermined constants c lt c 2 , and we must write 

w l : (w l - w 2 ) :w 2 = <p% : c, j& : c a yj. 

The constants c are now to be so determined that these two equations 
give actually only one equation between z and w . This is possible when 
and only when 

is an identity in z l and z 2 . Now this relation can be satisfied by definite 
constants c x and c 2 . A brief calculation shows that the identity 

must hold, so that the octahedral equation (3) becomes: 
(6) w l : (w l - w z ) : w 2 = $ : j& : 108 yj - 

This equation surely maps the points , 1 , oo respectively upon the 
centers of gravity of the faces, the midpoints of the edges, and the vertices 
of the octahedron, with the proper multiplicity, because the forms 90, %, y 
were so constructed. Furthermore, the twenty-four octahedron substi- 

1 See Ihosaeder, p. 54. 

Continuation: Setting up the Normal Equation. 

tutions (4) transform it into itself, for they transform the zeros of each 
of the forms <p,%, y into themselves and at the same time change 
each of the forms by a multiplicative factor. And calculation shows 
that these factors cancel when the quotients are formed. 

It only remains to show that equation (6) really maps each shaded or 
unshaded triangle of the z sphere conformally upon the rear or front w hemi- 
sphere. We know that the points 0, 1 , oo of the real w meridian corres- 
pond to the three vertuces of each of the triangles; but the equation 
has, moreover, twenty-four roots z for each value of w. Since these 
must distribute themselves among the twenty-four triangles, w can 
take a given value but once, at most, within a triangle. If we could 
only show that w remains real on the three sides of a triangle, we could 
then easily show that there is a one-to-one mapping of each side upon 
a segment of the real w meridian, and also a similar mapping of the 
entire interior of the triangle upon the corresponding hemisphere, one which 
is conformal without reversal of angles. You will be able to make these 
deductions yourselves by making use of the continuity and the analytic 
character of the function w (z) . I shall indicate the only noteworthy step 
of the proof, that of showing the reality of w upon the sides of the triangle. 

It is more convenient to prove this by showing that w is real 
upon all the great circles that arise in the octahedral subdivision. These 
are, first, the three mutually perpendicular circles which pass each 
through four of the six vertices of the octahedron (principal circles] 
full drawn in Fig. 50, p. 127), and, second, the six circles, corresponding 
to the altitudes of the faces, which bisect the angles of the principal 
circles (auxiliary circles] long dotted in Fig. 50). By means of the octa- 
hedron substitutions, one can transform every principal circle into any 
other and every auxiliary circle into any other. Hence it will suffice 
to show that the function w is real at every point on one principal and 
one auxiliary circle, since it must take the same values on the other 
circles. Now the meridian of real numbers z is one of the principal 
circles. By (6), the values on this circle are 

which are, of course, real, since y and y are real polynomials in z l and z 2 . 
Of the auxiliary circles let us select the one through and oo which makes 
an angle of 45 with the real meridian and on which z takes the values 


z = e 4 r , where r ranges through real values from oo to + - 
On this circle 2 4 = e ia r 4 = r 4 is real. Since by (5) only the fourth 
powers of z l and z 2 occur in (p Q and in the fourth power of y e , the last 
formula shows that w is real. 

This concludes the proof: Equation (6), in fact, maps the w hemisphere, 
or the Riemann surface over it, conformally upon that triangular subdivision 

-JIQ Algebra: Equations in the Field of Complex Quantities. 

of the z sphere which corresponds to the octahedron, and consequently we 
have in this case, as completely as in the earlier examples, a geometric 
control of the dependence which this equation sets up between z and w. 
The treatment of the tetrahedron and of the icosahedron proceeds 
according to the same plan. I shall give only the results. As before, 
these results are those obtained when the subdivision of the z sphere 
has the simplest possible position. The tetrahedral equation 1 is 

wi = K ~ ^2)^2 = fe - 2|^3*f*i + 4 s 

and the icosahedral equation* is 

w, : (w, - w 2 ) : z*> 2 - {- (zf + 2?) + 228 &V 2 - ****) - 494} 3 
: -{ (*?+*?) + 522 (zf^-^zf) - 10005 (? 

i.e., these equations map the w hemispheres conformally upon the shaded 
and the unshaded triangles of that subdivision of the z sphere which belongs 
to the tetrahedron and to the icosahedron respectively. 

5. Concerning the Solution of the Normal Equations 
Let us now consider somewhat the common properties of the equations 
which we have been discussing and which we shall call the normal 

Note, first of all, that the extremely simple nature of all our normal 
equations is due to the fact that they have exactly the same number of 
linear substitutions into themselves as is indicated by the degree, i.e., that 
all their roots are linear functions of a single one\ and, further, that we 
have, in the divisions of the sphere, a very obvious geometric picture of all 
w-Sphere: ^ e re ^ a ^ ons ^ a ^ comeup for consideration. Just how 

simple many things appear which are ordinarily 
quite complicated with equations of such high degree 
will be evident if I raise a certain question in con- 
nection with the icosahedral equation. 

Let a real value W Q be given, say on the segment 
(1 , oo) of the real w meridian (see Fig. 51). Let us 
inquire about the sixty roots z of the icosahedral 
equation when w W Q . Our theory of the mapping 
tells us at once that one of them must lie on a side of each of the sixty 
triangles on the z sphere which arise in the case of the icosahedron (drawn 
full in Fig. 49, P- 125). This supplies what one calls, in the theory of 

1 See Ikosaeder, p. 51, 60. 2 Loc. cit., p. 56, 60. 

Concerning the Solution of the Normal Equations. 

equations, the separation of the roots, usually a laborious task, which 
must precede the numerical calculation of the roots. The task is that of 
assigning separated intervals in each of which but one root lies. But we 
can also tell at once how many of the roots are real. If we take into 
account, namely, that the form of the icosahedral equation given above 
implies such a placing 1 of the icosahedron in the z sphere that the real 
meridian contains four vertices of each of the three sorts a,b,c, then it 
follows (see Fig. 48, p. 123, and Fig. 49, p. 119) that four full-drawn 
triangle sides lie on the real meridian, so that there are just four real 
roots. The same is true if w lies in one of the other two segments of the 
real w meridian, so that for every real w different from , 1 , <x> the icosa- 
hedral equation has four real and fifty-six imaginary roots] for w = 0, 
1 , oo there are also four different real roots, but they are repeated. 

I shall now say something about the actual numerical calculation of 
the roots of our normal equations. We have here again the great ad- 
vantage that we need to calculate but one root, because the others follow 
by linear substitutions. Let me remind you, however, that the numerical 
calculation of a root is really a problem of analysis, not of algebra, since 
it requires necessarily the application of infinite processes when the root 
to which one is approximating is irrational, as is the case in general. 

I shall go into details only for the simplest example of all, the pure 

w == z rt . 

Here I come again into immediate touch with school mathematics. For this 

n i 

equation, i. e., the calculation of yw, at least for the small values of n 
and for real values of w = r, is treated there also. The method of cal- 
culating square and cube root, as you learned it in school, depends, 
in essence, upon the following procedure. One determines the position 
which the radicand w = r has in the series of the squares or cubes, 
respectively, of the natural numbers 1, 2, 3,... Then, using the 
decimal notation, one makes the same trial with the tenths of the 
interval concerned, then with the hundreths, and so on.. In this way 
one can, of course, approximate with any desired degree of closeness. 
I should like to apply a more rational process, one in which we can 
admit not only arbitrary integral values of n but also arbitrary complex 

values of w . Since we need to determine only one solution of the equation, 

we shall seek, in particular, that value z = y w which lies within the 

angle 2 n\n laid off on the axis of real numbers. Generalizating the ele- 
mentary method mentioned above, we begin by dividing this angle into 
v equal parts (v = 5 in Fig. 52), and by drawing circles intersecting the 
dividing rays by circles which have the origin as common center and 

1 See Ikosaeder, p. 55. 

132 Algebra: Equations in the Field of Complex Quantities. 

whose radii are measured by the numbers r = 1 , 2 , 3 , . . . In this way, 
after choosing v, we find all the points 

2i7t k /k = Q, 1, 2, . . ., v \ 
z = r-e" n v ( /=1 ,2,3,... 

marked within the angular space, and we can at once mark in the 
w plane the corresponding w values 

w = z n = r n e v . 

These will be the corners of a corresponding network (see Fig. 53) 
covering the entire w plane and consisting of circles with radii l w , 2 n , 
3 n , . . . together with rays inclined to the real axis at angles of 0, 2 n\v , 


Fig. 53. 

(^ 1) 2Ji[v. Let the given value of w lie either within or 
on the contour of one of the meshes of this lattice, and suppose that W Q 

n. - 

is the lattice corner nearest to it. We know a value Z Q of ]/ze> is a corner 
of the lattice in the z plane; hence the value we are seeking will be 

We expand the right side by the binomial theorem, which we may con- 
sider known, inasmuch as we are now, in reality, in the domain of 

We can decide at once as to the convergence of this series if we look 
upon it as the Taylor's development of the analytic function ^w and apply 
the theorem that it converges within the circle which has W Q as 

n . 

center and which passes through the nearest singular point. Since }w 
has only and oo as singular points, our development will converge if, 
and only if, w lies within that circle about W Q which passes through the 
origin, and we can always bring this about by starting, in the z plane, 
with a similar lattice which may have smaller meshes, if necessary. 
But in order that the convergence should be good, i.e., in order that the series 

Uniformization of the Normal Irrationalities. 


should be adapted to numerical calculation (w WQ)/W Q must be sufficiently 
small. This can always be effected by a further reduction of the lattice. 
This is really a very usable method for the actual calculation of numerical 

Now is it worthy of remark that the numerical solution of the remaining 
normal equations of the regular solids is not essentially more difficult, but 
I shall omit the proof. If we apply, namely, the same method to our 
normal equations, starting from the mapping upon the w sphere of two 
neighboring triangles, there will appear, in place of the binomial series, 
certain other series that are well known in analysis and are well adapted 
to practical use, called the hypergeometric series. In the year 1877 
I set up 1 this series numerically. 

6. Uniformization of the Normal Irrationalities by Means of 
Transcendental Functions 

I shall now discuss another method of solving our normal equations 
which is characterized by the systematic employment of transcendental 
functions. Instead of proceeding, in each individual case, with series 
developments in the neighborhod of a known solution, we try to re- 
present, once for all, the whole set of number pairs (w , z) which satisfy 
the equation, as single-valued analytic functions of an auxiliary variable: 
or, as we say, to uniformize the irrationalities defined by the equation. 
If we can succeed by using only functions which can easily be tabulated, 
or of which one already has, perhaps, numerical tables, one can obtain 
the numerical solution of the equation without farther calculation. I am 
the more willing to discuss this connection with transcendental functions 
because it sometimes plays a part in school instruction, where it still 
often has a hazy, almost mysterious, aspect. The reason for this is that 
one is still clinging to traditional imperfect conceptions, although the 
modern theory of functions of a complex variable has provided perfect 

I shall apply these general suggestions first to the pure equation. 
Even in the schools, we always use logarithms in calculating the positive 
solution of z n = r , for real positive values of r. We write the equation 
in the form z = e l sr/n , where logr stands for the positive principal 
value. The logarithmic tables supply first log r, and then, conversely, z 
is the number that corresponds to log r/n. Moreover, we ordinarily use 
10 as base instead of e. This solution can be extended immediately to 
complex values. We satisfy the equation 

z n = w , 

[ l Weiteve Untersuchungen uber das Ikosaeder, Mathematische Annalen, vol. 12, 
p. 515. See also Klein, F., Gesammelte Mathematische Abhandlungen, vol. 2, p. 331 
et seq.] 

Algebra: Equations in the Field of Complex Quantities. 

by putting x equal to the general complex logarithm, log w , after which 
we obtain w and z actually as single-valued analytic functions of x : 


w = e x , z = e n 

In view of the many-valuedness of x = log w, which we shall study 
later in detail, one obtains here for the same w precisely n values of z . 
We call x the uniformizing variable. 

Since the tables contain only the real logarithms of real numbers, 
we are apparently unable to read off immediately the value of the given 
solution. But by the aid of a simple property of logarithms, we can 
reduce the calculation to the use of trigonometric tables which are accessible 
to everybody. If we put 

w = u -f- iv = 

then the first factor, as a positive real number, has a real logarithm, 
the second, as a number of absolute value 1 , a pure imaginary logarithm 
i<p (i.e., the second factor is equal to e iv ), and we obtain <p from 
the equation 

u v 


This gives x = log w = log | }u* + v 2 \ +iq>, and the root of the equation 
is therefore 

-- loglV^+^l Liy 
z = e n = e n *e n 

i. e., we have 

X1 v n log I Vu* + v* I / qj , . . m\ 

(b) z = yu + iv = e n (cos + tsm--j. 

Since <p is determined only to within multiples of 2 n , this formula 
supplies all the n roots. With the aid of ordinary logarithmic and 
trigonometric tables, we can now get first q> from (a) und then z from (b). 
We have obtained this "trigonometric solution" from the logarithms of 
complex numbers in an entirely natural way. However, if we assume 
that these are not known and try to develop this trigonometric solution, 
as is done in the schools, it must appear as something entirely foreign 
and unintelligible. 

Occasionally it becomes necessary to find roots of numbers that are 
not real. Thus, in school instruction, such roots must be found in the 
so called Cardan's solution of the cubic equation about which I should 
like to interpolate here a few remarks. If this equation is given in the 
reduced form 
(1) x* + px q = 0, 

Uniformization of the Normal Irrationalities. 

then the formula of Cardan states that its three roots x lt x 2 , # 3 are 
contained in the expression 



Since every cube root is three valued, this expression has, all told, 
nine values, in general all different; among these, x lt x%, x$ are deter- 
mined by the condition that the product of the two cube roots employed 
each time is p/^ . If we replace the coefficients p , q in the well known 
manner by their expressions as symmetric functions of x l9 x z , x 3 , and 
if we note that the coefficient of x 2 vanishes, that is, x l + x% + x$ = 0, 
we get 

q I p (#1 #2) (^2 ^3) v^s ^i) 

T 27 ~~ 7 108 ' 

that is, the radicand of the square root is, to within a negative factor, 
the discriminant of the equation. This shows at once that it is negative 
when all three roots are real, but positive when one root is real and the other 
two conjugate imaginary. It is precisely in the apparently simplest case 
of the cubic equation, namely when all the roots are real, that the 
formula of Cardan requires the extraction of the square root of a nega- 
tive number, and hence of the cube root of an imaginary number. 

This passage through the complex must have seemed something 
quite impossible to the mediaeval algebraists at a time when one was 
still far removed from a theory of complex numbers, 250 years before 
Gauss gave his interpretation of them in the plane! One talked of the 
"Casus irreducibilis" of the cubic equation and said that the Cardan 
formula failed here to give a reasonable usable solution. When it was 
discovered later that it was possible, precisely in this case, to establish 
a simple relation between the cubic equation and the trisection of an 
angle, and to get in this way a real "trigonometric solution* ' in place 
of the defective Cardan formula, it was believed that something new 
Had been discovered which had no connection with the old formula. 
Unfortunately this is the position taken occasionally even today in 
elementary instruction. 

In opposition to this view, I should like to insist here emphatically 
that this trigonometric solution is nothing else than the application, in 
calculating the roots of complex radicands, of the process which we have 
just discussed. It is obtained therefore in a perfectly natural way in 
this case, where the cube root has a complex radicand, if we transform 
the Cardan formula, for numerical calculation, in the same convenient 
way that one pursues in school for the case of the real radicand. In fact, 
let us suppose 


Algebra: Equations in the Field of Complex Quantities. 

where p must be negative if q is real. If we then write the first cube 
root in (2) in the form 


-+.-I !/--! 

We note that its absolute value value (as positive cube root of the 
value V p*/27 of the radicand) is equal to | V pft ; but since the 
product of this by the second cube root is equal to p/3 , that second 
cube root must be the conjugate complex of this, and the sum of the 
two, i.e., the solution of the cubic equation, is simply twice the real 
part, that is, 


Now let us apply the general procedure of p. 134. We write the 
radicand of the cube root, after separating out its absolute value, in 
the form 


I/ ?2 

t> B 




IF 4 27 



]/ ^ 

I * 

I/ ^ 3 

V 27 

r 27 

and determine 

an angle 

9? from the equations 



r 4 27 


TJJ" ' 

L 1 




Then, since the positive cube root of |V p*/27\ is |V 
root takes the form 

, our cube 

and hence, remembering that (p is determinate only to within multiples 
of 2^, we obtain 

' COS J 

But this is the usual form of the trigonometric solution. 

I should like to take this opportunity to make a remark about the 
expression "casus irreducibilis" . "Irreducible" is used here in a sense 
entirely different from the one in use today and which we shall often 
use in these lectures. In the sense here used it implies that the solution 
of the cubic equation cannot be reduced to the cube roots of real numbers. 
This is not in the least the modern meaning of the word. You see how 
the unfortunate use of words, together with the general fear of cBmplex 
numbers, has created at least the possibility for a good deal of misunder- 

Uniformization of Normal Irrationalities. 


standing in just this field. I hope that my words may serve as a preven- 
tive, at least among you. 

Let us now inquire briefly about uniformization by means of trans- 
cendental functions in the case of the remaining normal irrationalities. 
In the dihedral equation 

z n + = 2w 

we put simply 

w = cos <f . 

De Moivre's formula shows that the equation is then satisfied by 

w . . . w 

z = cos + & sin . 

n n 

Since all values of 9? + 2 k n and of 2 k n q> gi ve the same value of w 
this formula gives, in fact, for every w, 2n values of z, which we can 

(p + 2kn . . . cp 4- 2kn .. 

z = cos 5^ t sin^ ------ . (k = 0, 1 , 2, . . ., n - 1) 

In the case of the equations of the octahedron, tetrahedron, and 
icosahedron these "elementary" transcendental functions do not suffice. 
However, we can obtain the corresponding solution by means of elliptic 
modular functions. Although one may not consider this solution as 
belonging to elementary methematics, I should, nevertheless, like to 
give, at least, the formulas 1 which relate to the icosahedron. They are, 
namely, closely related to the solution of the general equation of degree 
five by means of elliptic functions, to which allusion is always made 
in textbooks and about which I shall have something to say later by 
way of explanation. The icosahedral equation had the form (see pp. 130, 

Now we identify w with the absolute invariant / from the theory of 
elliptic functions and think of / as a function of the period quotient 
w = oVojg (in Jacobi's notation i K'jK), i.e., we set 

t . 

zf(e lf o> 2 ) 

where g 2 and A are certain transcendental forms of dimension 4 and 
12, respectively, in o^ and co 2 , which play an important role. If we 
introduce the usual abbreviation of Jacobi 


q = e iato = e TK 

t 1 See Mathematische Annalen, vol. 14 (1878/79), p. Ill et seq., or Klein, 
Gesammelte Abhandlungen, vol. 3, p. 13 et seq., also Ikosaeder, p. 131.] 

Algebra: Equations in the Field of Complex Quantities. 

the roots z of the icosahedral equation will be given by the following 
quotients of ft functions 


If we take into account that co as a function of w, coming from the 
first equation, is infinitely many- valued, then this formula yields in 
fact all sixty roots of the icosahedral equation for a given w. 

7. Solution in Terms of Radicals 

There is one question in the theory of the normal equations which 
I have not yet touched, namely, whether or not our normal equations 
yield algebraically anything that is essentially new; and whether or 
not they can be resolved into one another or, in particular, into a sequence 
of pure equations. In other words, is it possible to build up the solution* 
of these equations in terms of w by means of a finite number of radical 
signs, one above another? 

So far as the equations of the dihedron, tetrahedron, and octahedron 
are concerned, it is easy to show, by means of algebraic theory, that 
they can be reduced, in fact, to pure equations. It will be sufficient 
if I give the details here for the dihedral equation only: 

z n + ^ = 2w. 

If we set: 

* W = C, 
the equation goes over into 

t 2 - 2w+ \ =0. 
It follows from this that 

= w /w 2 1 , 
and consequently 

-which is the desired solution by means of radicals. 

On the other hand, however, the icosahedral equation does not admit 
such a solution by means of radicals, so that this equation defines an 
essentially new algebraic function. I am going to give you a particularly 
graphic proof of this, which I have recently published (Mathematische 
Annalen, Vol. 61 [1905]), and which follows from consideration of the 
familiar function theoretic construction of the icosahedral function z (w) . 
For this purpose I shall need the following theorem, due to Abel, a 
proof of which you will find in every treatise on algebra: // the solution 
of an algebraic equation can be expressed as a sequence of radicals, then 
every radical of the sequence can be expressed as a rational function of 
the n roots of the given equation. 

Solution in Terms of Radicals. 

Let us now apply this theorem to the icosahedral equation. If we 
assume its root z can be expressed as a sequence of roots of rational 
functions of the coefficients, i.e., of rational functions of w t then every 
radical in the sequence is a rational function of the sixty roots: 

R (z l , z 2 , . . . , 2 60 ) . 

(We shall show that this leads to a contradiction.) In the first place, 
we can replace this expression by a rational function R (z) of z alone 
since all the roots can be derived from any one of them by a linear 
substitution. Let us now convert this R (z) into a function of w by 
writing for z the sixty-valued icosahedral function z (w) , and consider 
the result. Since every circuit in the w plane which returns z to its 
initial value must of necessity return R (z) also to its initial value, it 
follows that +R[z(w)] can have branch points only at the positions 
w = 0, 1 , oo (where z (w) has branch points), and the number of leaves 
of the Riemann surface for R [z (w)] which are cyclically connected at 
each of these positions must be a divisor of the corresponding number 
belonging to z (w). We know that this number is 3, 2, 5 at the three 
positions, respectively. Hence every rational function R (z) of an icosa- 
hedral root, and consequently every radical which appears in the assumed 
solution, considered as function of w, can have branch points, if at all, 
only a,tw = Q,w = \,w=oo. If branching occurs, then there must 
be three leaves connected at w = 0, two at w = \ , and five at w = oo, 
since 3,2,5 have no divisor other than 1 . 

We shall now see that this result leads to a contradiction. To this 
end let us examine the innermost radical which appears in our hypothe- 
tically assumed expression for z (w). Its radicand must be a rational 
function P (w) . We can assume that the index of the radical is a prime 
number p, since we could otherwise build it up out of radicals with 
prime indices. Moreover P (w) cannot be the ^?-th power of a rational 
function Q (w) of w, for if it were, our radical would be superfluous, 
and we could direct our attention to the next really essential radical. 

Let us now see what kind of branchings the function y P (w) can 
have. For this purpose it will be convenient to write it in the homo- 
geneous form 

where g and h are forms of the same dimension in the variables w l , w% 
(w = Wi/w 2 ) . According to the fundamental theorem of algebra we 
can separate g and h, into linear factors and write 


140 Algebra: Equations in the Field of Complex Quantities. 

since the numerator and the denominator are of the same degree. Not 
all the exponents a, ft, ...,<*', $' . . . can be divisible by p, since P 
would then be a perfect p-th power. On the other hand, a + /? + ... 
' /?' . . . is equal to zero, and is therefore divisible by p. 
Consequently at least two of these numbers are not divisible by p. 
It follows that the zeros of both the corresponding linear factors must 

be branch points of /P(z0), at each of which p leaves are cyclically 
connected. But herein lies the contradiction of the previous theorem, 

which, of course, must be equally valid for V P (w) . For we enumerated 
at that time all possible branch points, and we found among them no 
two at which the same number of leaves were connected. Our assumption 
is therefore not tenable, and the icosahedral equation cannot be solved 
by radicals. 

This proof depends essentially upon the fact that the numbers 3 , 2 , 5 
which are characteristic for the icosahedron have no common divisor. 
When such a common divisor appears, as in the case of the numbers 3 , 
2, 4 of the octahedron, it is at once possible to have rational functions 
R [z (w)] which exhibit the same kind of branching at two points, e.g., 
one in which two leaves are connected at 1 and at oo, and these can 
then be really represented as roots of a rational function P (w) . It is 
in this way that the solution by means of radicals comes about in the 
case of the octahedron and tetrahedron (with the numbers 3*2,3), 
and of the dihedron (2,2,w). 

I should like to show you here how slightly the language used in 
wide mathematical circles keeps pace with knowledge. The word "root" 
is used today nearly everywhere in two senses: once for the solution 
of any algebraic equation, and, secondly, in particular, for the solution 
of a pure equation. The latter use, of course, dates from a time when 
only pure equations were studied. Today it is, if not actually harmful, 
at least rather inconvenient. Thus it seems almost a contradiction to 
say that the "roots' ' of an equation cannot be expressed by means of 
radical signs. But there is another form of expression which has lingered 
on from the beginnings of algebra and which is a more serious source of 
misunderstanding, namely, that algebraic equations are said to be "not 
algebraically solvable", if they cannot be solved in terms of radicals 
i. e. if they cannot be reduced to pure equations. This use is in immediate 
contradiction with the modern meaning of the word "algebraic". Today 
we say that an equation can be solved algebraically when we can reduce 
it to a chain of simplest algebraic equations in which one controls the 
dependence of the solutions upon the parameters, the relation of the 
different roots to one another, etc. as completely as one does in the 
case of the pure equation. It is not at all necessary that these equations 
should be pure equations. In this sense we may say that the icosahedral 

Reduction of General Equations to Normal Equations. \^\ 

equation can be solved algebraically, for our discussion shows that we 
can construct its theory in a manner that meets all the demands men- 
tioned above. The fact that this equation cannot be solved by radicals 
lends it special interest by suggesting it as an appropriate normal 
equation to which one might try to reduce, (i. e., completely solve) 
still other equations which are in the old sense algebraically unsolvable. 
The last remark leads us to the last section of this chapter, in which 
we shall try to get a general view of such reductions. 

8. Reduction of General Equations to Normal Equations 

It turns out, namely, that the following reductions are possible: 
The general equation of the third degree to the dihedral equation forn = ^\ 
The general equation of the fourth degree to the tetrahedral or to the 
octahedral equation] 

The general equation of the fifth degree to the icosahedral equation. 
This result is the most recent triumph of the theory of the regular 
bodies which have always played such an important r61e since the 
beginning of mathematical history, and which have a decisive influence 
in the most widely separated fields of modern mathematics. 

In order to show you the meaning of my general assertion I shall 
go somewhat more into details for the equation of degree three, without, 
however, fully proving the formulas. We again take the cubic equation 
in the reduced form 

(1) # 3 + px q = 0. 

Denoting solutions by x lt x 2 , x$, we try to set up a rational function z 
of them which undergoes the six linear substitutions of the dihedron for 
n = 3 when we interchange the Xi in all six possible ways. The values 
that z should take on are 

z, ez, e z z, , ," I where = < 
z z z \ 

It is easily seen that 

(2} z Xl + sx * + ' 

\ ' V _L r-2^ l_ 

satisfies these conditions. The dihedral function z* + \/z 3 of this quantity 
must remain unaltered by all the interchanges of the Xk, since the 
six linear substitutions of the z leave it unchanged. Hence, by a well 
known theorem of algebra, it must be a rational function of the co- 
efficients of (1). A calculation shows that 

(3) * + . = -27^-2. 

Conversely, if we solve this dihedral equation, and if z is one of its 
roots, we can express the three values x l9 x%, x 3 rationally in terms of 

142 Algebra: Equations in the Field of Complex Quantities. 

z, p, and q by means of (2) and the well known relations 

Doing this, we find 

X *=P 

v __3? 

Thus, as soon as the dihedral equation (3) has been solved, the formulas 
(4) give at once the solution of the cubic (1). 

In the same way we may reduce the general equations of the fourth 
and fifth degrees. The equations would be, of course, somewhat longer, 
but not more difficult in principle. The only new thing would be that 
the parameter w of the normal equation, which was expressed above 

rationally in the coefficients of the equation \2w= 27^2], 

would now contain square roots. You will find this theory for the 
equation of degree five given fully in the second part of my lectures 
on the icosahedron. Not only are the formulas calculated, but also the 
essential reasons for the appearance of the equations are explained. 
Finally, let me say a word about the relation of this development 
to the usual presentation of the theory of equations of the third, fourth, 
and fifth degree. In the first place, we can obtain the usual solutions 
of the cubic and biquadratic from our formulas by appropriate reduc- 
tions, if we use the solutions of the equations of the dihedron, octahedron, 
and tetrahedron in terms of radicals. In the case of equations of degree 
five, most of the textbooks confine themselves unfortunately to the 
establishment of the negative result that the equation cannot be solved 
by radicals, to which is then added the vague hint that the solution 
is possible by elliptic functions, to be exact one should say elliptic 
modular functions. I take exception to this procedure because it ex- 
hibits a one-sided contrast and hinders rather than promotes a real 
understanding of the situation. In view of the preceding survey, using 
first algebraic and then analytic language, we may say: 

1 . The general equation of the fifth degree cannot be reduced, indeed, 
to pure equations, but it is possible to reduce it to the icosahedral equation 
as the simplest normal equation. This is the real problem of its algebraic 

2. The icosahedral equation, on the other hand, can be solved by elliptic 
modular functions. For purposes of numerical calculation, this is the 
full analog of the solution of pure equations by means of logarithms. 

Reduction of General Equations to Normal Equations. 

This supplies the complete solution of the problem of the equation 
of fifth degree. Remember that when the usual road does not lead to 
success, one should not be content with this determination of impossi- 
bility, but should bestir oneself to find a new and more promising route. 
Mathematical thought, as such, has no end. If someone says, to you that 
mathematical reasoning cannot be carried beyond a certain point, you 
may be sure that the really interesting problem begins precisely there. 

In conclusion, it might be remarked that these theories do not stop 
with equations of degree five. On the contrary, one can set up analogous 
developments for equations of the sixth and higher degrees if one will 
only make use of the higher-dimensional analogs of the regular bodies. 
If you are interested in this, you might read my article 1 Ober die Auf- 
losung der allgemeinen Gleichung funften und sechsten Grades*. In con- 
nection with this article the problem was successfully attacked by 
P. Gordan 2 and A. B. Coble 3 . The investigation is somewhat simplified 
in the latter memoir 4 . 

1 Journal fiir Mathematik,.vol. 129 (1905), p. 151 ; and Mathematische Annalen, 
vol. 61 (1905), P- 50. 

* Concerning the solution of the general equation of fifth and of sixth degree. 

2 Mathematische Annalen, vol. 61 (1905), p. 50; and vol. 68 (1910), p. 1. 

3 Mathematische Annalen, vol. 70 (1911), p- 337- 

4 See also Klein, F., Gesammelte Mathematische Abhandlungen, vol. 2, 
p. 502-503. 

Part Three 


During this second half of the semester we shall select certain chapters 
in analysis which are important from our standpoint and we shall 
discuss them as we did arithmetic and algebra. The most important 
thing for us to discuss will be the elementary transcendental functions, 
i. e. logarithmic and exponential functions and trigonometric functions, 
since they play an important part in school instruction. Let us begin 
with the first. 

I. Logarithmic and Exponential Functions 

Let me recall briefly the familiar curriculum of the school, and the 
continuation of it to the point at which the so called algebraic analysis 

1. Systematic Account of Algebraic Analysis 

One starts with powers of the form a = b c , where the exponent c 
is a positive integer, and extends the notion step by step for negative 
integral values of c, then for fractional values of c, and finally, if cir- 
cumstances warrant it, to irrational values of c. In this process the 
concept of root appears as that of a particular power. Without going 
into the details of involution, I will only recall the rule for multiplication 

which reduces the multiplication of two numbers to the addition of 
exponents. The possibility of this reduction, which, as you know, is 
fundamental for logarithmic calculation, lies in the fact that the fun- 
damental laws for multiplication and addition are so largely identical, 
that both operations, namely, are commutative as well associative. 
The operation inverse to that of raising to a power yields the 
logarithm. The quantity c is called the logarithm of a to the base 6: 


At this point a number of essential difficulties appear which are 
usually passed over without any attempt at explanation. For this reason 

Systematic Account of Algebraic Analysis. 

I shall try to be especially clear at this point. For the sake of convenience 
we shall write x and y instead of a and c , inasmuch as we wish to study 
the mutual dependence of these two numbers. Our fundamental equa- 
tions then become 

x = b y , y = log*. 


Let us first of all notice that b is always assumed to be positive. If b 
were negative, x would be alternately positive and negative for integral 
values of y, and would even include imaginary values for fractional 
values of y , so that the totality of number pairs (x , y) would not give 
a continuous curve. But even with b > one cannot get along without 
making stipulations that appear to be quite arbitrary. For if y is 
rational, say y=m/n f where m and n are integers prime to each other, 


x = b mfn is, as you know, defined to be y b m and it has accordingly n 
values, of which, for even values of n, we should have two to deal with 
even if we confined ourselves to real numbers. It is customary to 
stipulate that x shall always be the positive root, the so-called principal 

If you will permit me to use, somewhat prematurely, the familiar 
graph of the logarithm y = logx (Fig. 54), you will see that neither 
the above stipulation nor its suit- 
ableness is by any means self-evident. "* V * S 4^ > 
If y traverses the dense set of rational 
values, the corresponding points whose 
abscissas are the positive principal 
values x = b y constitute a dense set 
on our curve. If, now, when the de- 
nominator n of y is even, we should Fig. 54. 
mark the points which correspond to 

negative values of x, we have a set of points which would be, one might 
say, only half so dense, but nevertheless dense on the curve which is the 
reflection in the y axis of our curve [y = log (x)]. If we now admit 
all real, including irrational, values of y , it is certainly not immediately 
clear why the principal values which we have been marking on the right 
now constitute a continuous curve and whether or not the set of negative 
values which we have marked on the left is similarly raised to a con- 
tinuum. We shall see later that this can be made clear only with the 
profounder resources of function theory, an aid which is not at the com- 
mand of the elementary student. For this reason, one does not attempt 
in the schools to give a complete exposition. One adopts rather an 
authoritative convention, which is quite convincing to the pupils, 
namely that one must take b > and must select the positive principal 
values of x, that everything else is prohibited. Then the theorem follows, 

Klein, Elementary Mathematics. 10 

146 Analysis: Logarithmic and Exponential Functions. 

of course, that the logarithm is a single- valued function defined only for 
a positive argument. 

Once the theory is carried to this point, the logarithmic tables are 
put into the hands of the pupil and he must learn to use them in practical 
calculation. There may still be some schools in my school days this 
was the rule where little or nothing is said as to how these tables 
are made. That was despicable utilitarianism which is scornful of every 
higher principle of instruction, and which we must surely and severly 
condemn. Today, however, the calculation of logarithms is probably 
discussed in the majority of cases, and in many schools indeed the 
theory of natural logarithms and the development into series is taught 
for this purpose. 

As for the first of these, the base of the system of natural logarithms 
is, as you know, the number 

* = lim(l + } H = 2.7182818 

n=<x) \ n i 

This definition of e is usually, in imitation of the French models, placed 
at the very beginning in the great text books of analysis, and entirely 
unmotivated, whereby the really valuable element is missed, the one 
which mediates the understanding, namely, an explanation why pre- 
cisely this remarkable limit is used as base and why the resulting 
logarithms are called natural. Likewise the development into series is 
often introduced with equal abruptness. There is a formal assumption 
of the development 

log (1 + *) = a + a v % + a 2 * 2 H , 

the coefficients a , a l9 . . ., are calculated by means of the known pro- 
perties of logarithms, and perhaps the convergence is shown for | # | < 1 > 
But again there is no explanation as to why one would ever even suspect 
the possibility of a series development in the case of a function of such 
arbitrary composition as is the logarithm according to the school de- 

2. The Historical Development of the Theory 

If we wish to find all the fundamental connections whose absence 
we have noted, and to ascertain the deeper reasons why those apparently 
arbitrary conventions must lead to a reasonable result, in short, if we 
wish really to press forward to a full understanding of the theory of 
logarithms, it will be best to follow the historical development in its 
broad outlines. You will see that it by no means corresponds' to the 
practice mentioned above, but rather that this practice is, so to speak, 
a projection of that development from a most unfavorable standpoint. 

We shall mention first a German mathematician of the sixteenth 
century, the Swabian, Michael Stifel, whose Arithmetica Integra appeared 

The Historical Development of the Theory. 

in Niirnberg in 1544. This was the time of the first beginnings of our 
present algebra, a year before the appearance, also in Niirnberg, of the 
book by Cardanus, which we have mentioned. I can show you this 
book, as well as most of those which I shall mention later, thanks to 
our unusually complete university library. You will find that it uses, 
for the first time, operations with powers where the exponents are any 
rational numbers, and, in particular, emphasizes the rule for multi- 
plication. Indeed, Stifel gives, in a sense, the very first logarithmic 
table (see p. 250) which, to be sure, is quite rudimentary. It contains 
only the integers from 3 to 6 as exponents of 2 , along with the corres- 
ponding powers |- to 64. Stifel appears to have appreciated the signi- 
ficance of the development of which we have here the beginning. He 
declares, namely, that one might devote an entire book to these re- 
markable number relations. 

But in order to make logarithms really available for practical calcula- 
tion Stifel lacked still an important device, namely, decimal fractions ; 
and it was only when these became common property, after 1600, that 
the possibility arose of constructing real logarithmic tables. The first 
tables were due to the Scotchman Napier (or Neper), who lived 1550 1 61 7. 
They appeared in 1614, in Edinburgh, under the title Mirifici logarith- 
morum canonis descriptio, and the enthusiasm which they aroused is 
evidenced by the verses with which different authors in their prefaces 
sang the virtues of logarithms. However, Napier's method for calculating 
logarithms was not published until 1619, after his death, as Mirifici 
logarithmorum canonis construct 1 . 

The Swiss, Jobst Biirgi (15521632), had calculated a table in- 
dependently of Napier, which did not appear, however, until 1620, in 
Prag, under the title Arithmetische und geometrische Progresstabuln. We, 
in Gottingen, should have a peculiar interest in Biirgi, as one of our 
countrymen, since he lived for a long time in Cassel. In general, Cassel, 
particularly the old observatory there, has been of importance for the 
development of arithmetic, astronomy, and of optics prior to the 
discovery of infinitesimal calculus, just as Hannover became important 
later as the home of Leibniz. Thus our immediate neighborhood was 
historically significant for our science long before this university was 

It is very instructive to follow the train of thought of Napier and 
Biirgi. Both start from values of x = b y for integral values of y and 
seek an arrangement whereby the numbers % shall be as close together 
as possible. Their object was to find for every number #, as nearly as 
possible, a logarithm y . This is achieved today, in school, by considering 
fractional values of y, as we saw before. But Napier and Biirgi, with the 

1 Lugduni 1620. There is a later edition in phototype. (Paris 1895-) 


148 Analysis: Logarithmic and Exponential Functions. 

intuition of genius, avoided the difficulties which thus present themselves 
by grasping the thing by the smooth handle. They had, namely, the 
simple and happy thought of choosing the base b close to one, when, 
in fact, the successive integral powers of b are close to one another. 
Biirgi takes 

b = 1.0001, 
while Napier selects a value less than one, but still closer to it: 

6 = 1- 0.0000001 = 0.9999999- 

The reason for this departure by Napier from the method of today is 
that he had in mind the application to trigonometric calculation, where 
one has to do primarily with logarithms of proper fractions (sine and 
cosine) and these are negative for b > 1 but positive for b < 1 . But 
with both investigators the chief thing was that they made use only 
of integral powers of this b and so avoided, completely, the many valued- 
ness which embarrassed us above. 

Let us now calculate, in the system of Biirgi, the powers for two 
neighboring exponents, y and y + 1 : 

x + Ax = 

By subtraction, then, we have 

A x = (1.0001)" (1.0001 1) = ^ 

or, writing Ay for the differences, 1, of the values of the exponent: 

(la) 7* = - 

v ' Ax x 

We have thus obtained a difference equation for the Biirgi logarithms, 
one which Biirgi himself used directly in the calculation of his tables. 
After he had determined the oc corresponding to a y he obtained the 
following % belonging to y + 1 by the addition of #/10 4 . In the same 
way it follows that the logarithms of Napier satisfy the difference 

db) = - 

v ' Ax x 

In order to see the close relationship between the two systems, we 
need only write for y on the one hand y/10 4 , on the other hand y/10 7 , i.e., 
we need only displace the decimal point in the logarithm. If we denote 
the new numbers so obtained simply by y, we shall have in each case 
a series of numbers which satisfy the difference equation 


v ' 


and in which the values of y proceed by steps of 0.0001 in the one case 
and of 0.0000001 in the other. 

The Historical Development of the Theory. 


If, for the sake of convenience, we now make use of the graph of 
the continuous exponential curve (we ought really to obtain it as the 
result of our discussion) we shall have a tangible representation of the 
points which correspond to the number series of Napier and of Biirgi. 
These points will be the corners of a stairway inscribed in one of the 
two exponential curves 

(3) x = (1.0001) 10000 *, and x = (0.9999999) 10000000y , 

respectively, where the risers have the constant value Ay =0.0001 and 
Ay 0.0000001 in the two systems, respectively (see Fig. 55). 

We can get another geometric interpretation in which we do not 
need to presuppose the exponential curve, which will rather point out 
the natural way to obtain that curve, if we 
replace the difference equation (2) by a summa- 
tion equation, that is, if we integrate it, in a sense : 

(4) ,-V" 

Fig. 55- 

During this summation increases disconti- 
nuously, from unity on, by such steps that the 
corresponding A r\ = J/ is always constant and 

equal to 10 ~ 4 and 10 ~ 7 respectively, so that A = f/10 4 and /10 7 , in 
the two cases. With the last step f attains the value x. Once can easily 
give geometric expression to this procedure. For this purpose let us 
draw the hyperbola rj = 1/f in an 
f ?7 plane (see Fig. 56) and, begin- 
ning at ^ = 1 , construct succes- 
sively on the | axis the points that 
are given by the law of progression 
A | = /10 4 (confining ourselves to 
the Biirgi formulation). The rect- 
angle of altitude 1/f erected upon 
each of the intervals so ob- 
tained will have the constant area 
/! i/| = i/io 4 . The Biirgi logarithm will then be, according to (4), 
the 10 4 -fold sum of all these rectangles inscribed in the hyperbola and 
lying between 1 and x. A similar result is obtained for the logarithm 
of Napier. 

Proceeding from this last representation, one is led immediately to 
the natural logarithm if, instead of the sum of the rectangles, one takes 
the area under the hyperbola itself between the ordinates f = 1 and 
. = x (shaded in the figure). This finds expression in the well-known 

150 Analysis: Logarithmic and Exponential Functions. 

This was, in fact, the historical way, and the decisive step was taken 
about 1650, when analytic geometry had become the common possession 
of mathematicians and the infinitesimal calculus was achieving the 
quadrature of known curves. 

If we desire to use this definition of the natural logarithm as our 
starting point, we must, of course, convince ourselves that it possesses 
the fundamental property of replacing the multiplication of numbers 
by the addition of logarithms ; or, in modern terms, we must show that 
the function 

defined thus by means of the area under the hyperbola, has the simple 
addition theorem 

/(*i) +/(* 2 ) = /(*i'*2) 

In fact, if we vary x l and x 2> then, according to the definition of an 
integral, the increments of the two sides dx^x^ + dx 2 /x 2 and 
d (x l x 2 )/(x l # 2 ) are equal. Consequently / (x^ + / (x 2 ) and / (x x z ) 
can differ only by a constant, and this turns out to be zero when 
we put x l = 1 (since / (1) = 0). 

If we wish to determine the "base" of the logarithms obtained in 
this way, we need only notice that the transition from the series of 
rectangles to the area under the hyperbola can be made by changing 
the increment A = f /10 4 to A = /n and allowing n to become infinite. 
This is the same thing as replacing the Biirgi sequence x (1.0001) 10000z/ 
by x = (1 + \\n) ny , where ny becomes infinite through integral values. 
According to the general definition of a power, this amounts to saying 
that x is the y-th power of (1 + \jn) n . Accordingly it seems plausible 
to say that the base is lim (1 + \/n) n , the very limit which is ordinarily 


assumed at the start as the definition of e. It is interesting to note, 
moreover, that Biirgi's base (1.0001) 10000 = 2.718146 coincides with e 
to three decimal places. 

Let us now examine the historical development of the theory of 
the logarithm after Napier and Biirgi. First of all I shall make the 
following statements. 

1. Mercator, whom we have already met in these pages (see p. 81) 
was one of the first to make use of the definition of the logarithm by 
means of the area of the hyperbola. In his book Logarithmotechnica 
of 1668, as well as in articles in the Philosophical Transactions of the 
London Royal Society in 1667 and 1668, he shows, by means of the 
same argument which I have just given you in modern terms, that 

dSIS differs from the common logarithm with the base 10, which 

was already the base used in calculations, only by a constant factor, 

The Historical Development of the Theory. 

the so called modulus of the system of logarithms. Moreover he had 
already introduced 1 the name "natural logarithm" or "hyperbolic 
logarithm". But the greatest achievement of Mercator was the setting 
up of the power series for the logarithm, which he obtained (essentially, 
at least) from the integral representation by dividing out and integrating 
term by term. I mentioned this to you (p. 81) as an epochmaking 
advance in mathematics. 

2. In that same connection, I told you also that Newton had taken 
up these ideas of Mercator 1 s and had enriched them with two important 
results, namely, the general binomial theorem and the method for the 
reversion of series. This last appeared in a work of Newton's youth 
De analysi per aequationes numero terminorum infinitas which appeared 
late in print but which from 1669 on was distributed in manuscript 
form 2 . In this 3 Newton derives the exponential series 

for the first time by reverting Mercator' s series for y = log A;. This 
yields, as the number whose natural logarithm y = \ 

_ l 1 1 

6 " 1 + 1 ! + 27 + 3"! + ' " ' 

and it is now easy, with the aid of the functional equation for the 
logarithm, to show that, for every real rational y, # is one of the values 
of e y , and in fact the positive value, in the sense of the customary 
definition of power. We shall go into this more in detail later on. The 
function y = log % thus turns out to be precisely what one would 
call the logarithm of x to the base e , according to the ordinary definition, 
in which e is defined by means of the series and not as lim (1 + \/n) n . 


3. Brook Taylor could follow a more convenient path in deriving 
the exponential series, after he had devised the general series-development 
which bears his name, which appeared in his work Methodus Incremen- 
torum* and of which we shall have much to say later on. He could 
then use the relation 

d\ogx _ J_ 
dx ~~ ~x ' 

which is implied in the integral definition of the logarithm, infer from 
it the inverse relation 

de * y 

-= = e y 

1 Philosophical Transactions of the Royal Society of London, vol. 3 (1668), 
p. 761. 

2 Newton, I.: Opuscula, Tome I, op. 1, Lausanne 1744. Appeared first in 1711. 

3 Loc. cit., p. 20. 

4 London, 1715. 

152 Analysis: Logarithmic and Exponential Functions. 

and so write down at once the exponential series as a special case of 
his general series. 

We have already seen (p. 82) how this productive period was followed 
by the period of criticism, I should almost like to say the period of 
moral despair, in which every effort was directed toward placing the 
new results upon a sound basis and in separating out what was false. 
Let us now see what attitude was taken toward the exponential function 
and the logarithm in the books of Euler and Lagrange, which tended 
in this new direction. 

We shall begin with Euler 's Introductio in analysin infinitorum 1 . 
Let me, first of all, praise the extraordinary and admirable analytic 
skill which Euler shows in all his developments, noting, however, at 
the same time, that he shows no trace of the rigor which is demanded 

At the head of his developments Euler places the binomial theorem 

in which the exponent I is assumed to be an integer. Now integral 
exponents are not considered in the Introductio. This development is 
specialized for the expression 

/. . \ \ n V 

in which ny is integral. He then allows n to become infinite, applies 
this limit process to each term of the series, thinks of e as defined by 
lim (1 + l/w) n , and so obtains the exponential series 

To be sure, Euler is not in the least concerned here as to whether or not the 
individual steps in this process are rigorous, in the modern sense; in 
particular, whether the sum of the limits of the separate terms of the 
series is really the limit of the sum of the terms, or not. Nowthis derivation 
of the exponential has been, as you know, a model for numerous text- 
books on infinitesimal calculus, although, as time went on, the different 
steps have been more and more elaborated and their legitimacy put to 
the test of rigor. You will see how influential Euler' s work has been 
for the entire course of these things if you recall that the use of the 
letter e for that important number is due to him. "Ponamus autem 
brevitatis gratia pro numero hoc 2.71828 . . . constanter litteram e" , 
as he writes on page 90. 

1 Lausanne, 1748, Caput VII, p. 85 et seq. Translation by Maser, Berlin 1885, 
p. 70. [See also vol. VIII (1923) of Euler's Works, edited by F. Rudio, A. Krazer, 
and P. Stackel.] 

The Historical Development of the Theory. >jci 

I might add that Euler immediately follows this with an entirely 
analogous derivation of the series for the sine and cosine. For this pur- 
pose he starts with the development of sin (p in powers of sin (fpfn) 
and lets n become infinite. This is nothing else than a limit process 
applied to the binomial theorem, as is evident if one obtains the power 
series in question from De Moivre's formula: 

, / <P , - <p\ n i <p\ n ( w\ n 

cos<p + % sm<p = Jcos-^ + % sm-J-j - ^cos J (1 + t tg-Jj . 

Let us now consider Lagrange' s Theorie des fonctions analytiques 1 . 
Again it is to be noted that questions of convergence are treated, at 
most, only incidentally. I have already stated (p 83) that Lagrange 
considers only those functions that are given by power series, and defines 
their differential quotients formally by means of the derived power series. 
Consequently the Taylor's series 

is for him simply the result of a formal reordering of the series for 
/ (x + h) proceeding originally according to powers of x + h. Of course, 
if one wishes then to apply this series to a given function, one ought 
really to show in advance that this function is analytic, i.e., that it can 
be developed into a power series. 

Lagrange begins with the investigation of the function / (x) = x n , 
for rational n , and determines /' (x) as the coefficient of h in the expansion 
of (x + h) n , the first two terms of which he thinks of as calculated. 
Then, by the same law, he obtains at once /" (x) , /'" (x) , . . . , and the 
binomial expansion of (x + h) n appears as a special case of Taylor's 
series for / (x + h) . Moreover, let me note expressly that Lagrange does 
not give special consideration to the case of irrational exponents, but 
rather looks upon it as obviously settled when he has considered all 
rational values. It is interesting to contemplate this fact, since it is 
upon the rigorous justification of precisely this sort of transition that 
the greatest importance is laid today. 

Lagrange uses these results in a similar treatment of the function 
/ (x) = (1 + 6)*. By recording the binomial series for (1 + b) x+h he 
finds, namely, f (x) as the coefficient of h, then determines /"(#), 
/'" (x) , . . . according to the same law, and forms, finally, the Taylor 
series for / (x + h) = (1 + b} x+h . He is then in possession, for h = 0, 
of the desired exponential series. 

I should like now to finish this brief historical sketch, in which 
I have, of course, mentioned only names of the very first rank, by in- 
dicating what essentially new turns came with the nineteenth century. 

1 Paris, 1797, Reprinted in Lagrange, CEuvres, vol. 4. Paris 1881. Compare 
especially chapter 3, p. 34 et seq. 

154 Analysis: Logarithmic and Exponential Functions, 

1 . At the head of this list I should place the precise ideas concerning 
the convergence of infinite series and other infinite processes. Gauss 
takes precedence here with his Abhandlung uber die hypergeometrische 
Reihe* in 1812 (Disquisitiones generates circa seriem infinitam 
1 + [(a b)/(i c)] x + - ) l . After him comes Abel with his memoir on the 
binomial series in 1826 (Untersuchungen uber die Reihe 1 + (m/\)x + 2 ), 
while Cauchy, in the early twenties in his Cours d' Analyse 9 undertook, for 
the first time, a general discussion of the convergence of series. The result 
of these investigations, for the series which we have under consideration, 
is that all the earlier developments are sometimes correct, although the 
rigorous proofs are very complicated. For the detailed consideration of 
such proofs, in modern form, I refer you again toBurkhardt'sAlgebraische 
Analysis or to Weber- Wellstein. 

2. Although we shall have occasion to talk about it in detail later, 
I must mention here the final foundation by Cauchy of the infinitesimal 
calculus. By means of it the theory of the logarithm, which we discussed 
above as taking its start at the hands of Biirgi and Napier in the seven- 
teenth century, was established with full mathematical exactness. 

3. Finally, we must mention the rise of that theory which is in- 
dispensable to a complete understanding of the logarithmic and ex- 
ponential functions, namely, the theory of functions of a complex 
argument, often called, briefly, function thedry. Gauss was the first 
to have a complete view of the foundations of this theory, even though 
he published little or nothing concerning it. In a letter to Bessel, dated 
December 18, 1811, but published much later 4 , he sketches and explains 

with admirable clearness the significance of the integral / dz[z in the 

complex plane, in so far as it is an infinitely many-valued function. 
The fame of having also created independently the complex function 
theory and of having made it known to the mathematical world belongs, 
however, to Cauchy. 

The result of these developments, insofar as it concerns our special 
subject, might be briefly stated as follows: The introduction of the 
logarithm by means of the quadrature of the hyperbola is the equal in 
rigor of any other method, wheteas it surpasses all others, as we have 
seen, in simplicity and clearness. 

* Memoir on the hypergeometric series. 

1 Commentationes societatis regiae Gottingiensis recentiores, vol. 11 (1813), 
No. l, pp. 1 46. Werke vol. 3, pp. 123 162. German translation by Simon, 
Berlin 1888. 

2 Journal ftir Mathematik, vol. 1 (1826), pp. 311 339- Ostwalds Klassiker 
No. 71. 

3 Premiere Partie, Analyse Algdbrique. Paris 1821. = CEuvres, 2nd series, vol. 3, 
Paris, 1897. German translation by Itzigsohn. Berlin 1885- 

4 Briefwechsel zwischen Gauss unct Bessel, edited by Auwers. Berlin 1880; 
or Gauss Werke, vol. 8 (1900), p. 90. 

The Theory of Logarithms in the Schools. 155 

3. The Theory of Logarithms in the Schools 

It is remarkable that this modern development has passed over the 
schools without having, for the most part, the slightest effect on the 
instruction, an evil to which I have often alluded. The teacher manages 
to get along still with the cumbersome algebraic analysis, in spite of 
its difficulties and imperfections, and avoids the smooth infinitesimal 
calculus, although the eighteenth century shyness toward it has long 
lost all point. The reason for this probably lies in the fact that mathe- 
matical instruction in the schools and the onward march of investigation 
lost all touch with each other after the beginning of the nineteenth 
century. And this is the more remarkable since the specific training 
of future teachers of mathematics dates from the early decades of that 
century. I called attention in the preface to this discontinuity, which 
was of long standing, and which resisted every reform of the school 
tradition : In the schools, namely, one cared little whether and how the 
given theorems were extended at the university and one was therefore satis- 
fied often with definitions which were perhaps sufficient for the present, 
but which failed to meet later demands. In a word, Euler remained 
the standard for the schools. And conversely, the university frequently 
takes little trouble to make connection with what has been given in 
the schools, but builds up its own system, sometimes dismissing this 
or that with brief consideration and with the inappropriate remark: 
"You had this at school". 

In view of this, it is interesting to note that thpse university teachers 
who give lectures to wider circles, e.g. to students of natural science 
and technology, have, of their own accord, adopted a method of intro- 
ducing the logarithm which is quite similar to the one which I am 
recommending. Let me mention here, in particular, Scheffer's Lehrbuch 
der Mathematik fur Studierende der Naturwissenschaften und Technik* 1 . 
You will find there in chapters six and seven a very detailed theory 
of the logarithm and the exponential function, which coincides entirely 
with our plan and which is followed in chapter eight by a similar theory 
of the trigonometric functions. I urge you to make the acquaintance 
of this book. It is very appropriate for teachers, for whom it is designed, 
in that the material is presented fully, in readable form, and adapted 
to the comprehension even of the less gifted. Note, too, the great 
pedagogic skill of Scheffers when he (to cite one example) continually 
draws attention to the small number of formulas in the theory of 
logarithms that one needs to know by heart, provided the subject is 
once understood; for one can then easily look them up when they are 
needed. In this way he encourages the reader to persevere in face of 

* Textbook of Mathematics for Students of Natural Science and Technology. 
1 Leipzig, 1905; fifth ed. 1921. 

-JJ6 Analysis: Logarithmic and Exponential Functions. 

the great mass of new material. I call your attention also to the fact 
that although Scheffers takes it for granted that the subject has been 
studied in school, he nevertheless develops it here in detail, on the 
assumption that most of what was learned in school has been forgotten. 
In spite of this, it does not occur to Scheffers to make proposals for a 
reform of instruction in the schools, as I am doing. 

I should like to outline briefly once more 
my plan for introducing the logarithm into 
the schools in this simple and natural way. 
rj~i<>lL The first principle is that the proper source 
~~~~ from which to bring in new functions is the 

quadrature of known curves. This corre- 
sponds, as I have shown, not only to the 
historical situation but also to the procedure 
in the higher fields of mathematics, e. g., in 
elliptic functions. Following this principle 

one would start with the hyperbola 77 = \ / and define the logarithm 
of x as the area under this curve between the ordinates = 1 and = x 
(see Fig. 57). If the end ordinate is allowed to vary, it is easy to see 
how the area changes with and hence to draw approximately the 
curve r] = log . 

In order now to obtain simply the functional equation of the logarithm 
we can start with the relation 

/ x d^ f cx dl; 
L T = Jo T' 

which is obtained by applying the transformation c = ' to the variable 
of integration. This means that the area between the ordinates 1 and x 
is the same as that between the ordinates c and ex which are c times 
as far from the origin. We can make this clear geometrically by ob- 
serving that the area remains the same when we slide it along the 
axis under the curve provided we stretch the width in the same ratio 
as we shrink the height. From this the addition theorem follows at once : 

Ji Ji A Jxi Ji 

I wish very much that some one would give this plan a practical 
test in the schools. Just how it should be carried out in detail must, of 
course, be decided by the experienced school man. In the Meran school 
curriculum we did not quite venture to propose this as the standard 

4. The Standpoint of Function Theory 

Let us, finally, see how the modern theory of functions disposes of 
the logarithm. We shall find that all the difficulties which we met in 

The Standpoint of Function Theory. 

our earlier discussion will be fully cleared away. From now on we shall 
use, instead of y and x> the complex variables w = u + iv and z = x 
+ iy. Then 

1. The logarithm is defined by means of the integral 

(1) w- 

where the path of integration is any curve in the f plane joining ? = 1 
to f = *. 

2. The integral has infinitely many values according as the path 
of integration encircles the origin 0,1,2,... times, so that log z is 
an infinitely-many-valued function. 

One definite value, the principal -Plane: 

value [log z] , is determined if we 

slit the plane along the negative real 

axis and agree that the path of 

integration shall not cross this cut. 

It still remains arbitrary, of course, 

whether we shall choose to reach 

the negative real values from above Fi gi 5 s. 

or from below. According to the 

decision on this point the logarithm has + n i or ni for its imaginary 

part. The general value of the logarithm is obtained from the principal 

value by the addition of an arbitrary multiple of 2in\ 

(2) log* = [log z] + 2kni, (k = 0, 1 , 2, . . .) . 

3. It follows from the integral definition of w = logz that the 
inverse function z = f (w) satisfies the differential equation 

From this we can at once write down the power series for / 

it \ * i w . w * . w * i 
, = /() = !+_ + _+_+.... 

Since this series converges for every finite w, we can infer that the 
inverse function is a single-valued function which can be singular only 
for w = oo, i.e., that it is an integral transcendental function. 

4. The addition theorem for the logarithm is derived from the 
integral definition, just as for real variables. From it we obtain for 
the inverse function the equation 

(4) /K)-/K) = /(^i + ^). 

Similarly, it follows from (2) that 

(5) f(w + 2kni) = f (w), (k = 0, 1, 2, . . .) 

i.e., / (w) is a simply periodic function with the period 2 n i . 

1J8 Analysis: Logarithmic and Exponential Functions. 

5. If we put / (1) = e y it follows from (4) that for every rational 

n t 
value m/n of w the function / (w) will be one of the n values of y e m , as 

this expression is usually defined; that is 

We shall adopt the customary notation, and denote this one value of 
f(w) by e w = e m/n , so that e w is a well defined single-valued function, 
and indeed, the one given by equation (3). 

6. What sort of a function, then, shall we understand, in the most 
general sense, by the power b w with an arbitrary base 6? We must 
adopt such conventions, of course, that the formal rules for exponents 
are satisfied. In order then to establish a connection between b w and 
the function e w which we have just defined, let us put b equal to e ] Kb , 
where log b has the infinitely many values 

log b = [log 6] + 2 kni , (k = 0, 1 , 2, . . .) 
It follow then that 

b w = (e lo * b ) w = ^w- log ft e w[\oub] . 6 2kniv> 9 (& = 0, 1 

and this expression represents, for the different values of k, infinitely 
many functions which are completely unconnected. We have thus the 
remarkable result that the values of the general exponential expression 
b w , as these are obtained by the processes of raising to a power and 
extracting a root, do not belong at all to one coherent analytic function, 
but to infinitely many different functions of w , each of which is single- 

The values of these functions are, to be sure, related to each other 
in various ways. In particular they are all equal when w is an integer ; 
and there are only a finite number of different ones among them 
(namely, n) when w is a fraction mjn in its lowest terms. These n values 
are ^m/ioiog* . g2fct(m/n) for A = 0, 1 , . . . , n - 1 , that is, the n values of 

]/b m , as we should expect. 

7. It is only now that we can appreciate the inappropriateness of 
the traditional method which starts from involution and evolution and 
expects to arrive at a single- valued exponential function. It finds itself 
in an outright labyrinth in which it cannot possibly find its way by 
so called elementary means, especially since it restricts itself to real 
quantities. You will see this clearly if you will consider the situation 
when 6 is negative, with the aid of the illuminating results which we 
have just obtained. In this connection I merely remind you that we 
are only now in a position to understand the suitableness of the definition 
of the principal value (b > and b m/n > 0; see p. 145) which at the 

The Standpoint of Function Theory. 

time seemed arbitrary. It yields the values of one only of our infinitely 
many functions, namely those of the function 

[b w ] == M>[log6] 

On the other hand, if n is even, the negative real values of b m ^ n will 
constitute a set which is everywhere dense, but they belong to an 
entirely different one of our infinitely many functions, and cannot 
possible combine to form a continuous analytic curve. 

I should now like to add a few remarks of a more serious nature 
concerning the function theoretic nature of the logarithm. Since 
w = log 2 suffers an increment of 2ni every time z makes a circuit 
about z = 0, the corresponding Riemann surface of infinitely many 
sheets must have at z = a branch point of infinitely high order so 
that each circuit means a passage from one sheet into the next one. 
If one goes over to the Riemann sphere it is easy to see that z = <x> is 
another branch point of the same order and that there are no others. 
We can now make clear what one calls the uniformizing power of the 
logarithm of which we have already spoken in connection with the 
solution of certain algebraic equations (see p. 133 e * sc l-)- To fix ideas 
let us consider a rational power, z mfn . By reason of the relation 

m m . 


z n e n 

this power will be a single-valued function of w = log z . This is expressed 
by saying that it is uniformized by means of the logarithm. In order 
to understand this, let us think of the Riemann surface of z min as well 
as that of the logarithm, both spread over the z plane. This will have n 
sheets and its branch points will also be at 
2 = and z = oo, at each of which all the n 
sheets will be cyclically connected. If we 
now think of any closed path in the z plane 
(see Fig. 59) along which the logarithm returns 
to its initial value, which implies that its path 
on the infinitely many sheeted surface is also 
closed, it is easy to see that the image of this Fig. 59. 

path will likewise be closed when it is mapped 

upon the n sheeted surface. We infer from this geometric consideration 
that z m/n will always return to its initial value when log z does, and 
hence that it is a singlevalued function of log z. I am the more 
willing to give this brief explanation because we have here the sim- 
plest case of the principle of uniformization, which plays such an 
important part in modern function theory. 

We shall now try to make clearer the nature of the functional 
relation w = log z by considering the conformal mapping upon the 
w plane of the z plane and of the Riemann surface spread upon it. In 
order not to be obliged to go back too far, let us refrain from including 


Analysis: Logarithmic and Exponential Functions. 


w -Plane: 

the corresponding spheres within the scope of our deliberations, in spite 
of the fact that it would be preferable to do so. As before, we divide 
the z plane along the axis of reals into a shaded (upper) and a unshaded 
(lower) half plane. Each of these must have infinitely many images in 
the w plane, since log z is infinitely many valued, and all these images 
must lie in smooth connection with one another since the inverse function 
z = e w is one valued. This means that the w plane is divided into 
parallel strips of width n separated from one another by parallels to 
the real axis (see Fig. 60). These strips are to be alternately shaded 
and left blank (the first one above the real axis is shaded) and they 
represent, accordingly, alternate conformal maps of the upper and lower 

z half planes while the separating par- 
allels correspond to the parts of the real 
z axis. As to the correspondence in detail, 
I shall remark only that z always appro- 
aches when w , within a strip, tends to 
the left toward infinity, that z becomes 
infinite when w approaches infinity to 
the right, and that the inverse function e w 
has an essential singularity at w = oo. 
I must not omit here to draw attention 
to the connection between this represen- 
tation and the theorem of Picard, since that 
is one of the most interesting theorems of 
the newer function theory. Let z (w) be 
an integral transcendental function, that is, a function which has an 
essential singularity only at w = oo (e.g. e w ). The question is whether 
there can be values 2, and how many of them, which cannot be taken 
at any finite value of w, but which are approached as a limit when w 
becomes infinite in an appropriate way. The theorem of Picard states 
that a function in the neighborhood of an essential singularity can omit 
at most two different values; that an integral transcendental function, 
therefore, can omit, besides 2 = oo, (which it of necessity omits), at 
most one other value. e w is an example of a function which really 
omits one other value besides oo, namely 2 = 0. In each of the parallel 
strips of our division e w approaches each of these values but it assumes 
neither of them for any finite value of w . The function sin w is an example 
of a function which omits no value except 2 = oo. 

I should like to conclude this discussion by bringing up again a 
point which we have repeatedly touched and applying to it these 
geometric aids. I refer to the passage to the limit from the power to 
the exponential function which is given by the formula 

-- SiJt- 

Fig. 60. 

The Standpoint of Function Theory. 
If we put n w = v this takes the form 

Let us, before passing to the limit, consider the function 


whose function-theoretic behavior, as a power, is known to us. It has 

a critical point, at w = v and w = oo, where the base becomes 

and oo respectively, and it maps the / r half 

planes conformally upon sectors of the w w .pi ane: 

plane which have w = v as common vertex 

and the angular opening n/v (see Fig. 61). 

If v is not an integer this series of sectors 

can cover the w plane a finite or an infinite 

number of times, corresponding to the many 

valuedness of /,, . If now v becomes infinite, 

the vertex, v, of the sectors moves off Fig 61 

without limit to the left and it is clear that 

these sectors lying to the right of v go over into the parallel strips 

of the w plane which belong to the limit function e w . This explains 

geometrically the limit definition of e w . One can verify by calculation 

that the width of the sectors at w = goes over _ . 

w -Sphere? 

into the strip width n of the parallel division. 
But a doubt arises here. If v becomes infinite 
continuously, it passes through, not only integral 
but also rational and irrational values, for which 
the f v will be many valued and will correspond to 
many sheeted surfaces. How can these go over into 
the smooth plane which corresponds to the single- 
valued function e w ? If, for example, we allow v to 
approach infinity only through rational values having 
n for a denominator each /,, (w) will have an n sheeted Riemann surface. 
In order to follow the limit process, let us, for a moment, consider 
the w spKere. It is covered for each f v (w) with n sheets which are 
connected at the branch points v and oo. Let the branch cut lie 
along the minor meridian segment joining these points, as shown in 
Fig. 62. If now v approaches oo the branch points coincide and the 
branch cut disappears. Thus the bridge is destroyed that supplied the 
connection between the sheets, there emerge n separate sheets and, 
corresponding to them, n single-valued functions, of which only one is 
our e w . If we now allow v to vary through all real values, we shall have, 
in general, surfaces with infinitely many sheets whose connection is 
broken in the limit. The values on one leaf of each of these surfaces 

Klein, Elementary Mathematics. 1 1 

162 Analysis: The Goniometric Functions. 

converge toward the single-valued function e w t which is spread over the 
smooth sphere, while the sequences of values on the other sheets have, 
in general, no limit whatever. We thus have a complete explanation of 
the right complicated and wonderful passage to the limit from the many 
valued power to the single- valued exponential function. 

As a general moral of these last considerations we might say that a 
complete understanding of such problems is possible only when they 
are taken into the field of complex numbers. Is this, then, not a sufficient 
reason for teaching complex function theory in the schools? Max 
Simon, for one, has in fact supported similar demands. I hardly believe, 
however, that the average pupils, even in the highest class, can be 
carried so far, and I think, therefore, that we should abandon those 
aspects of method as to algebraic analysis in the schools which incline 
toward such considerations, in favor of the simple and natural way which 
we have developed above. I am, to be sure, all the more desirous that 
the teacher shall be in full possession of all the function-theoretic 
connections that come up here; for the teacher's knowledge should be 
far greater than that which he presents to his pupils. He must be 
familiar with the cliffs and the whirlpools in order to guide his pupils 
safely past them. 

After these detailed discussions we can now be briefer in the corres- 
ponding consideration of the goniometric functions. 

II. The Goniometric Functions 

Let me say, before beginning, that the name goniometric functions 
seems preferable to the customary name trigonometric functions, since 
trigonometry is but a particular application of these functions, which 
are of the greatest importance for mathematics as a whole. Their inverse 
functions are analogous to the logarithm, while they themselves are 
analogous to the exponential function. We shall call these inverse 
functions the cyclometric functions. 

1. Theory of the Goniometric Functions 

As a starting point for our theoretical considerations let me suggest 
the question as to the most appropriate way of introducing the gonio- 
metric functions in the schools. I think that here also it would be best 
to make use of our general principle of quadrature. The customary 
procedure, which begins with the measurement of the circular arc, does 
not seem to me to be so very obvious, and it lacks, above all, the ad- 
vantage of affording a simple and coherent control both of elementary 
and advanced fields. 

Again I shall make immediate use of analytic geometry. Let us 
start with the unit circle 

Theory of the Goniometric Functions. 


and consider the sector formed by the radii to the points A (x = 1 , 
y = o) and P (x, y) (see Fig. 63). In order to be in agreement with 
the usual notation, I shall denote the area of this sector by 90/2 . (Then 
the arc in the customary notation will be (p.) 

I shall define the goniometric functions sine and cosine of q> as the 
lengths of the coordinates x and y of the limiting point P of the sector 99/2 : 

x = cos 9?, 

y = sin 9? . 

The origin of this notation is not clear. The word "sinus" probably arose 

through an erroneous translation of an Arabic word into Latin. Since 

we did not start from the arc we cannot 

well designate the inverse functions, i. e., 

the double sector, as, a function of the 

coordinates, by using the customary terms 

arc sine and arc cosine, but it is natural 

by analogy to call <p\2 the "area 11 of the 

sine (or cosine) and to write 

<p = 2 area siny = arc siny , 
9? = 2 area cosx = arc cos x . 

Fig. 63. 

The following notation, used in England and in America is also quite 
appropriate : 

cp = sin - I y . 

= cos' 1 *, 

The further goniometric functions: 

, sin (p , 

tan w = - , ctno? = 

r cos (p ' T 

cos (jp 

- - r - 
sin (p 

(in the older trigonometry also secant and cosecant) are defined as 

simple rational combinations of the two fundamental functions. They 

are introduced only with a view 

to brevity in practical calcula- <**+- ^ 

tion and have for us no theo- 

retical significance. 

If we follow the coordinates 
of P with increasing 9? we can 
at once obtain qualitatively a 
representation of the cosine and 

sine curves in a rectangular coordinate system. They are the well 
known wave lines with a certain period 2 n (see Fig. 64), where n is 
defined as the area of the entire unit circle, instead of as usual, the 
length of the semi-circle. 

Let us now compare once more our introduction of the logarithm 
and the exponential function with these definitions. You will recall that 


Fig. 64. 

1(54 Analysis: The Goniometric Functions. 

our point of departure was a rectangular hyperbola referred to its 
asymptotes as axes. 

-17 = 1. 

The semi axis of this hyperbola is OA = ]/2 (see Fig. 65), whereas the 
circle had the radius 1 . Let us now consider the area of the strip between 
the fixed ordinate A A' (f = 1) and the variable ordinate PP'. If this 

is called , we may put log I , and the 
coordinates of P are expressed in terms of 
in the form 

You notice a certain analogy with the preceding 
discussion, but that the analogy fails in two 
respects. In the first place, is not a sector 
as it was before, and furthermore the two coor- 
dinates are now expressed rationally in terms of 
one function e fp , whereas, in the case of the circle, 

we had to introduce two functions, sine and cosine, to secure rational ex- 
pressions. We shall see however that this divergence can be easily resolved. 
Notice, in the first place, that the area of the triangle OP'P, namely 
1? = , i s independent of the position of P. In particular, then, 
it is the same as that of OA 'A . Therefore, if we add the latter triangle 
to and then subtract the former triangle from this sum, we see that 
can be defined as the area of a hyperbolic sector lying between a radius 
vector to the vertex A and one to a variable point P , jiist as in the case 
of the circle. There is still a difference in sign. Before, the arc AP, 
looked at from , was counterclockwise, whereas now it is clockwise. 
We can remove this difference by reflecting the hyperbola in OA , i.e., 
by interchanging and 77. We get then as coordinates of P 

Finally let us introduce the principal axes in place of the asymptotes 
as axes of reference, by turning Fig. 65 through 45 (after reflection 
in OA). If we call the new coordinates (X , Y), the equations of this 
transformation are 

f2 1/2" 

The equation of the hyperbola then becomes 

and the sector now has precisely the same position that sector 0/2 
had in the circle. The new coordinates of P as functions of may be 
written in the form 

j __ t + e 9 Y = e ~ e 

Theory of the Goniometric Functions. 


It remains only to reduce the entire figure in the ratio 1 : ]/2 in 

order to make the semi axis of the hyperbola 1 instead of the ]/2, as 
it was in the case of the circle. Then the sector in question has the area 
<p/2, in complete accord with the pre- 
ceding. If we call the new coordi- 
nates (x, y) again, they will be the 
following functions of 

</> I rh 

e ~r e 

/y . ._ 

; which satisfy the relation 

Fig. 66. 

which is the equation of a hyperbola. These functions are called hyper- 
bolic cosine and sine and are written in the form 

x = cosh = 

y = sinh = 

The final result, then, is that if we treat the circle and the rectangular 
hyperbola, each with semiaxis one, in literally the same way we obtain 
on the one hand the ordinary goniometric functions, on the other the 
hyperbolic functions, so that these functions correspond fully to one 

You know that these functions cosh and sinh can be used to ad- 
vantage in many cases. Nevertheless we have really taken a step back- 
ward here, so far as the treatment of the hyperbola is concerned. Whereas 
at first, the coordinates ( , rj) could be rationally expressed in terms 
of a single function e <t} ', it now requires two functions, which are connected 
by an algebraic relation (the equation of the hyperbola). It is natural, 
therefore to attempt a converse treatment for the goniometric functions, 
analogous to the original developments for the hyperbola. This is, in 
fact, quite easy if one does not object to the use of complex quantities, 
and it leads to the setting up of a single fundamental function in terms 
of which cos (p and sin y> can be expressed rationally, just as cosh < 
and sinh & are in terms of e*, and which is therefore entitled to play 
the chief role in the theory of the goniometric functions. 

To this end we introduce into the equation of the circle x 2 + y 2 = 1 
(where x = cos <p , y sin (p) the new coordinates 

which gives 


Analysis: The Goniometric Functions. 

The desired central function is now the second coordinate vf 9 just 
as it was above in the case of the hyperbola. If we denote it by / (99) 
we have, by virtue of the equations of transformation: 

i) = f(<p) = cos 99 + ishi99 , = TT-y = COS99 i sin 99 . 
From the last equations we get 

00599 = 

sin a? = - ~ . - = --- 
^ 2^ 


where we have complete analogy with the earlier relations between 
cosh #, sinh<, and e . If prominence is thus given, from the start, 
to the analogy between the circular and the hyperbolic functions, the 
great discovery of Euler that / (<p) = e ltf) is divested of the mystery 
that usually attaches to it. 

The question now arises whether we cannot effect a similar reduction 
of cos w and sin w to a single fundamental function, without leaving 

the real field. This is indeed possible 
if we look at our figures in the light 
of project! ve geometry. In the case 
of the hyperbola, in fact, we could 
define the coordinate r\, which sup- 
plied the fundamental function, as 
parameter in a pencil of parallels 
r] = constant. This means, projecti- 
vely, so far as the hyperbola is con- 
cerned, that we have a pencil of lines 
with its vertex on the hyperbola (in 
particular, here, at one of the infinitely 
distant points)* If, now, in the case of either circle or hyperbola we 
think of the parameter of any such pencil as a function of the area, 
we obtain likewise a fundamental function and one which involves only 
real quantities. 

Let us think now of the circle (Fig. 67) and the pencil through the 
point 5 (1,0) 

Fig. 67. 

where A is the parameter. On a former occasion (p. 45), we found as 
the coordinates of the intersection P of the circle and the ray correspond- 
ing to A, 

1 - A 2 . 2A 

x = cosy = y-j-jj- , y = sin 99 = j-^ . 

so that 

is, in fact, an appropriate real fundamental function. Moreover, since 
Z PSO = i POA , and POA = q>, it follows at once that JL = tan y/2. 

Theory of the Goniometric Functions. 167 

The one-valued representation of sin <p and cos y in terms of tan <p/2 
which appears in this way is often used in trigonometric calculations. 
The connection between A and the earlier fundamental function f(<p) 
appears from the last formula in the form 

; = _?__ == 1 /-/"* = 1 _ / a - 1 

* + 1 i "/ + /-i + 2 i /*'+"! + 2/ 

or conversely, 

,, , . l - A 2 

= * + i = 

The introduction of A amounts, then, simply to the determination of a 
linear fractional function of / (q>) which is real along the circumference 
of the unit circle. In this way the formulas turn out to be real but 
somewhat more complicated than by the immediate use of / (y) . 

Whether one is willing to give up the advantage of reality in the 
face of this disadvantage, depends, of course, upon how well the person 
concerned knows how to deal with complex quantities. It is noteworthy, 
in this connection, that physicists have long since gone over to the 
use of complex quantities, especially in optics, for example, as soon 
as they have to do with equations of vibration. Engineers, in particular 
electrical engineers with their vector diagrams, have recently been 
using complex quantities advantageously. We can say then that the 
use of complex quantities is at last beginning to spread, even though 
at present the great majority still prefer the restriction to real numbers. 

Passing on to a brief survey of the farther development of the theory 
of the goniometric functions, let us next consider certain fundamental 

1 . The addition theorem for sin <p is 

sin (cp -f- v) = sin 9? cos -^ + 

and there is a corresponding formula for cos (<p + ^) . These formulas 
appear to be more difficult than those for the exponential function, 
due, of course, to the fact that we are not dealing here with the true 
elementary function. This function, our / (9?) = cos q> + i sin <p, satis- 
fies the very simple relation 

v) = /(?)/(?). 

which is precisely the formula for e tp . 

2. It is easy now to obtain expressions for the functions of multiples 
of an angle and of parts of an angle. Of these I shall mention only the 
two formulas 

cos <p <p 

2 cos 2 

because they were of such importance in constructing the first trigono- 


Analysis: The Goniometric Functions. 

metric tables. An elegant expression for all these relations is given by 
De Moivre's formula 

f(n-<p) = 

where f(q>) = cos 9? + ism<p . 

De Moivre, who was a Frenchman, but who lived in London, and was 
in touch with Newton, published this formula in 1730 in his book 
Miscellanea analytica. 

3. From our original definition of y = sin <p, we can of course easily 
derive an integral representation for the inverse q> = sin" 1 )/. The area 
in Fig. 68, consisting of the sector <p/2 (A OP) of the unit circle, together 
with the triangle OP'P , is bounded by the axes, a parallel to the x axis 

at the distance y away, and the curve x = ]/l y 2 . Its area is there- 

ry i -- 

fore / Vl y*dy. Since the triangle has the area 

we have 

V 1 - y 2 dy - 

From this it follows by a simple transform- 
ation that 


cp = sin I y = 



Fig. 68. 

We could proceed now just as in the case of 
the logarithm, namely to develop the inte- 
grand by the binomial theorem, and then to integrate term by term, 
following Mercator. This would give us the power series for sin~ 1 y, 
from which, by inversion, we could get the sine series itself. This is 
the plan that Newton himself employed, as we have seen (p. 82). 

4. I prefer, however, to take the shorter way which Taylor's great 
discovery made possible. According to it one obtains from the above 
integral formula the differential quotient for the sine itself 

d y , o 

+ I/ A yllZ _ 

dqp d 

from which it follows that 


Taylor's theorem now gives 



= sin <p , 

. + 

> i I r | 



9> 4 

Theory of the Goniometric Functions. 


It is easy to see that these series converge for every finite <p , including 
complex values, and that sin (p and cos q> are therefore defined as single- 
valued integral transcendental functions in the entire complex plane. 
5. If we compare these series with the series for e v , we see that 
the fundamental function / (<p) satisfies the relation 

cos<p + i sin <p = e ir f . 

This result is unambiguous because sin <p and cos (p as well as e <p are 
single-valued integral functions. 

6. It remains only to describe the nature of the complex functions 
sin w , and cos w . We notice first that each of the inverse functions 
w = sin~~ l z and w = cos"" 1 z yields a Riemann surface with an infinite 


number of leaves and with branch points at +1, 1, oo. 

infinitely many branch points of the first order 

lie over 2 = 4-! and z = \, while two 

branch points of infinitely high order lie over 

z = oo . In order to follow better the course 

of the leaves in detail let us consider the divi- 

sion of the w plane into regions which corre- 

spond to the upper (shaded) and the lower 

(unshaded) z half planes. For z = cosw this 

division is brought about by the real axis and 

by the parallels to the imaginary axis through 

the points w = 0, n > 2 n, . . ., so that 

the resulting triangular regions (see Fig. 69), 

all extending to infinity, should be alternately 

shaded and unshaded. At the points w = , 

In fact, 

+ 7 

Fig. 69. 

ft , 4^, . . . (corresponding to z = +1), and at the points w = 

> - (corresponding to z = 1) , four of the triangles meet. These 
correspond to the four half leaves of the Riemann surface, which are 
connected at each of the corresponding branch points lying above 
z = -j-i f If w becomes infinite within any triangle, cos w approaches 
the value z = oo. The fact that there are two separate sets of infinitely 
many triangles each, all extending to infinity, corresponds to the situa- 
tion that on the Riemann surface there are two separate sets of infinitely 
many leaves connected at z = oo. For z = sin w the situation is 
analogous, except that the representation in the w plane is moved to 
the right by n/2. In these representations we find confirmation of my 
earlier remarks (p. 1 60) concerning the nature of the essential singularity 
at w = oo in its relation to the theorem of Picard. 

2. Trigonometric Tables 

After this brief survey of the theory of goniometric functions, 
I wish to discuss something that is of prime importance in practical work, 

-J70 Analysis: The Goniometric Functions. 

namely trigonometric tables. At the same time I shall talk about loga- 
rithmic tables, which I have thus far left in the background, for the 
reason that from the beginning up to the present time the tabulation 
of logarithms has gone hand in hand with that of trigonometric values. 
The way in which logarithmic tables have reached their present form 
is of extraordinary importance and interest for the mathematician in 
the schools as well as in the university. I cannot describe in detail here, 
of course, the long history of the development of such tables, but 
I shall endeavor, by citing a few of the most significant works, to give 
you a rough historical survey. Concerning other works, some of them 
of equally great importance, which would round out the story, I refer 
you to Tropfke or, so far as logarithmic tables are concerned, to the 
exhaustive account in Mehmke's Encyclopedia report on numerisches 
Rechnen (Enzyklopadie, I. F.), as well as to the French revision 1 of this 
report by d'Ocagne. 

I shall mention first the group of 

A. Purely Trigonometric Tables 

as they were developed before the invention of logarithms. Such tables 
existed in ancient times, the first of which follows. 

1 . The table of chords, by Ptolemy, which he compiled for astronomical 
purposes about 150 A. D. This is to be found in his work Megale Syn- 
taxis, in which he developed the astronomical system bearing his name, 
and of which we have here a modern edition 2 . This work has come to 
us, by way of the Arabs, under the much used title Almagest, which 
is probably a combination of the Arabic article "al" with a mutilated 
form of the Greek title. The table is constructed with thirty-minute 
intervals. It does not give directly the sine of the angle #, but the chord 
of its arc (i. e. 2 sin a/2) . The values of the chords are given in three 
place sexagesimal fractions, that is in the form 0/60 + 6/3600 + c/216000, 
where a,b,c are integers between and 59. The difficult thing for us, 
however, is that these a , b , c are written, of course, in Greek number- 
symbols, that is in combinations of Greek letters. The tables give also 
the values of the differences, which permit one to interpolate fcr minutes. 
In the calculation of his table, Ptolemy used, above all, the addition 
theorem for trigonometric functions, in the form of the theorem on the 
inscribed quadrilateral (Ptolemy's theorem). He used also the preceding 
formula for sin <x/2 (i.e., the extraction of square root, in addition to 
the rational operations), and he employed furthermore a process of inter- 

1 Encyclopedic des Sciences Mathematiques, edition francaise, I, 23. See also 
Cajori, F., History of Mathematics, 1919- Macmillan; and Smith, D. E., History 4 
of Mathematics, 1925- Ginn. 

2 Edited by Heiberg. 18981903. Leipzig. 

Trigonometric Tables. 

2. We advance now more than 1000 years to the time when tri- 
gonometric tables were first made in Europe. The first person who 
deserves mention is Regiomontanus (14361476), whose name was really 
Johannes Miiller, but who changed it into the latinized form of Konigs- 
berg, his birthplace. He calculated several trigonometric tables, in 
which one sees distinctly the transition from the sexagesimal to the pure 
decimal system. At that time no one thought of the trigonometric lines 
as fractions corresponding to the radius one, as we do now. The values 
were calculated for circles with very large radii, so that they appeared 
as integers. To be sure, these large numbers were themselves written 
as decimals, but in the choice of the radius one finds a persistent sug- 
gestion of the sexagesimal system. Thus, in the first table of Regio- 
montanus the radius is taken as 6000000, and not until he makes the 
second table does he choose a pure decimal 10000000 and establish 
complete accord with the decimal system. By the simple insertion of 
a decimal point, the numbers of this table become decimals of today. 
These tables of Regiomontanus were first published long after his death, 
in the work of his teacher G. Peurbach: Tractatus super propositiones 
Ptolemaei de sinubus et chordis 1 . Notice that this work, like so many 
other fundamental works in mathematics*, was printed in Niirnberg in 
the forties of the sixteenth century. Regiomontanus himself lived mostly 
in Niirnberg. 

3. I place before you now a work of the greatest general significance: 
De revolutionibus orbium coelestium* by Nic. Copernicus, the book in 
which the Copernican astronomical system is developed. Copernicus 
lived from 1473 to 1543 in Thorn, but this work appeared likewise in 
Niirnberg, two years after the publication of Regiomontanus 1 tables. 
Inasmuch as Copernicus never saw these tables, he was obliged to 
compute for himself the little table of sines which you find in his book 
and which was needed to work out his theory. 

4. These tables by no means met the needs of the astronomers, so 
that we see a pupil and friend of Copernicus attempting soon a much 
larger work. His name was Rhaticus, which again is a latinized form of 
the name, of his birthplace (Vorarlberg). He lived from 1514 to 1576, 
and was professor at Wittenberg. You must relate all these things to 
the general historical background of the time. Thus we are in the age 
of the Reformation when, as you know, Wittenberg and the free city 
Niirnberg were centers of intellectual life. Gradually, however, during 
the struggles of the Reformation, the center of gravity of the political 
and intellectual life moved away from the cities and toward the courts 
of the princes. Thus while everything heretofore had been printed in 

1 Norimbergae, 1541. 

* I have already mentioned Cardanus and Stifel and shall soon mention others. 

2 Norimbergae, 1543- 

172 Analysis: The Goniometric Functions. 

Niirnberg, the great tables of Rhaticus now appeared under the patronage 
of the Elector Palatine and bore therefore his name Opus Palatinum 1 . 
They were printed shortly after the death of Rhaticus. They were 
much more complete than the preceding tables, containing the values 
of the trigonometric lines to ten plaes at intervals of ten minutes, with, 
to be sure, a good many errors. 

5. A new edition of this table, very much improved, was published 
by Pitiscus of Griinberg in Silesia (1561 1613), chaplain of the Elector 
Palatine. This Thesaurus Mathematicus 2 , again printed under princely 
subsidy, contained the trigonometric numbers to fifteen places, at inter- 
vals of ten minutes. The work was essentially freer from errors than 
that of Rhaticus, and was more compendious. 

We must bear in mind that all these tables were constructed, in the 
main, with the aid solely of the half-angle formula, together with inter- 
polation, for at that time the infinite series for sin x and cos % did not 
exist. We can appreciate, then, the prodigious diligence and labor which 
is represented in these great works. 

B. Logarithmic-Trigonometric Tables 

These tables were succeeded immediately by the development of the 
second group, the logarithmic-trigonometric tables, and it is a re- 
markable coincidence, the irony of history, one might say, that a 
year after the tables of trigonometric lines had attained, with Pitiscus, 
a certain completeness, the first logarithms appeared and rendered these 
tables superfluous, in that from then on, instead of sine and cosine, 
one used their logarithms. I have already mentioned the first logarithmic 
tables, those of Napier. 

1. Mirifici Logarilhmorum Canonis Descriptio of Napier, in 1614. 
Napier had in mind, primarily, the facilitating of trigonometric cacula- 
tion. Consequently he did not give the logarithms of the natural num- 
bers, but only the seven-place logarithms of the trigonometric lines, at 
intervals of one minute. 

2. The actual construction of logarithmic tables in their present 
form is due mainly to the Englishman Henry Briggs (15564630) who 
was in touch with Napier. He recognized the great advantage that 
logarithms with base ten would have for practical calculation, since they 
would fit our decimal system better, and he introduced this base instead 
of that of Napier as early as 1617 in his Logarithmorum Chilias Prima, 
giving us the "artificial" or common logarithms which bear his name. 
In order to calculate these logarithms, Briggs devised a series of inter- 
esting methods which permitted the determination of each logarithm as 
accurately as one chose. Briggs' second considerable book bore the title 

1 Heidelbergae, 1596. 2 Francofurtii, 1613. 

Trigonometric Tables. \j<i 

Arithmetica logarithmica 1 . In it he tabulates the logarithms of the 
natural numbers themselves instead of those of the angle ratios, as Napier 
had done. To be sure, Briggs never finished his calculations. He gave 
the logarithms of the integers only from 1 to 20000 and 90000 to 100000, 
but to fourteen places. It is remarkable that precisely the oldest tables 
give the most places, whereas now we are content, for most purposes, 
with very few places. I shall come back to this later. Briggs also compiled 
the common logarithms of the trigonometric lines to ten places with 
ten jninute intervals in his Trigonometria Britannica-. 

3. The gap in Briggs' table was filled by the Dutchman Adrian 
Vlacq, mathematician, printer, and dealer in books, who lived inGouda 
near Ley den. He issued a second edition of Briggs' book 3 , which con- 
tained the logarithms of all integers from \ to 100000 but only to ten 
places. We may consider this as the source of all our current tables 
of logarithms of natural numbers. 

Concerning the further development of tables, I can mention here 
only in a general way the points in which advances were made in later 
years as compared with the above mentioned early beginnings. 

a) The first essential advance was in the theory. The logarithmic 
series furnished, namely, an extremely useful new method for the calcula- 
tion of logarithms. The compilers of the first tables knew nothing about 
these series. As we have seen, Napier calculated his logarithms by means 
of the difference equation, that is, by successive addition of A x/x, with 
the further aid of interpolation. The important device of square root 
extraction appeared with Briggs. He made use of the fact, which was 
mentioned moreover by Napier in his Constructio (see p. 147), that one 
knows log y<z b = \ (log a + log b) as soon as one knows the logarithms 
of a and b. It is probable that Vlacq also calculated in this way. 

b) Essential progress was made by a more suitable arrangement in 
printing the tables, whereby it was made possible to combine more 
material, in a clearer way, in a smaller space. 

c) Above all, the correctness of the tables, was considerably increased 
by a careful check of the older ones, thereby eliminating numerous 
errors, especially in the last figures. 

Among the large number of tables which thus appeared, I shall 
mention only the most famous one. 

4. This is the Thesaurus Logarithmorum Completus (Vollstandige 
Sammlung grosserer logarithmisch-trigonometrischer Tafeln*), by the 
Austrian artillery officer Vega, which appeared in Leipzig in 1 794. The 
original is rare, but a photostatic reprint appeared in Florence in 1896. 

1 Londini, 1624. 2 Goudae, 1633- 

3 Briggs, H., Arithmetica Logarithmica. Editio secunda aucta per Adr. Vlacq, 
Goudae, 1628. 

* Complete collection of larger logarithmic trigonometric tables. 

174 Analysis: The Goniometric Functions. 

The Thesaurus contains ten place logarithms of the natural numbers, 
and of the trigonometric lines, in an arrangement that has since become 
typical. Thus you find there, e.g., the small difference tables for facili- 
tating interpolation. 

If we come down now to the nineteenth century, we notice. a far 
reaching popularization of logarithms, due partly to the fact that they 
were introduced into the schools in the twenties, but also to the fact 
that they found more and more application in physical and technical 
practice. At the same time we find a reduction in the number of places. 
For the needs of the schools, as well as those of technical practice, were 
better met by tables which were not too bulky, especially since three 
or four places were sufficient for the requisite accuracy in nearly all 
practical cases. To be sure, we still had, in my school days, seven- 
place tables, the reason assigned being that the pupils would obtain in 
this way an impression of the "majesty of numbers' 1 . Our minds today 
are in general more utilitarian, and we use throughout two, three, or at 
most five-place tables. I shall show you today three modern tables, 
selected at random. One is a handy little four place table by Schubert 1 . 
In it you will find all manner of devices, such as printing in two colors, 
repetition above and below, on every page, of guiding quantities, and 
the like, in order to exclude misunderstanding. The second is a modern 
American table by Huntington 2 , which is still more cunningly arranged, 
where, e.g., the leaves are provided with projections and indentations 
to enable one to turn up at once the desired page. Finally, I am showing 
you a slide-rule, which, as you know, is nothing else than a three-place 
logarithmic table in the very convenient form of a mechanical calculator. 
You are all familiar, certainly, with this instrument, which every engineer 
nowadays has with him constantly. 

We have riot yet reached the end of the development, but we can 
see pretty clearly what its further direction will be. Of late, the cal- 
culating machine, of which I talked earlier (see p. 17 et seq.), has been 
coming into extensive use, and it makes logarithmic tables superfluous, 
since it permits a much more rapid and reliable direct multiplication. 
At present, however, this machine is so expensive that only large offices 
can afford it. When it has become considerably cheaper, a new phase 
of numerical calculation will be inaugurated. So far as goniometry is 
concerned, the old tables of Pitiscus, which became old fashioned so 
soon after birth, will then come into their own ; for they supply directly 
the trigonometric ratios with which the calculating machine can operate 
at once, thus avoiding the use of logarithms. 

[ l Now Schubert -Haussner, Vierstellige Tafeln und Gegentafeln, Sammlung 
Goschen, Leipzig, 191 7-] 

2 Huntington, C. V., Four- Place Tables. Abridged edition, Cambridge, Mas- 
sachusetts. 1907. 

Applications of Goniometric Functions. 175 

3. Applications of Goniometric Functions 

It remains for me now to give you a survey of the application of gonio- 
metric functions. I shall consider three fields 

A. Trigonometry, which, indeed, furnished the occasion for inventing 
the goniometric functions. 

B. Mechanics, where, in particular, the theory of small oscillations 
offers a wide field for applications. 

C. Representation of periodic functions by means of trigonometric series, 
which, as is well known, plays an important part in the greatest variety 
of problems. 

Let us turn at once to the first subject. 

A. Trigonometry, in particular, spherical trigonometry 

We are in the presence here of a very old science, which was in full 
flower in ancient Egypt, where it was encouraged by the needs of two 
important sciences. Geodesy required the theory of the plane triangle, 
and astronomy needed that of the spherical triangle. For the history 
of astronomy we have the voluminous monograph in A. v. Braun- 
muhl's Vorlesungen fiber Geschichte der Trigonometric 1 . On the practical 
side of trigonometry the most informative book is E. Hammer's: Lehr- 
buch der ebenen und sphdrischen Trigonometric 2 ', on the theoretical side, 
the second volume of the work I have often mentioned, the Enzyklopadie 
der Elementarmathematik of Weber- Wellstein. 

Within the limits of these lectures I cannot, of course, develop 
systematically the whole subject of trigonometry. That would be a 
matter for special study. Furthermore, practical trigonometry is given 
full consideration here in Gottingen in the regular lectures on geodesy 
and spherical astronomy. I should prefer to talk to you exclusively 
about a very interesting chapter of theoretical trigonometry which, in 
spite of its great age, cannot be regarded as closed, and which, on the 
contrary, contains many still unsolved problems and questions, of relati- 
vely elementary character, whose study would, I think, be rewarding. 
I refer to spherical trigonometry. You will find this subject very fully 
consideredin Weber-Wellstein, where importance is given to the thoughts 
which Study developed in his fundamental work Spharische Trigono- 
metric, orthogonale Substitutionen und elliptische Funktionen 3 . I shall try 
to give you a survey of all the theories that belong here and to call 
your attention to the questions which are still unanswered. 

The elementary notion of a spherical triangle hardly needs explana- 

1 Two volumes. Leipzig, 1900 and 1903- 

2 Stuttgart, 1906. [Fifth edition, 1923-] 

3 Abhandlungen der Mathematisch-physikalischen Klasse der Koniglich 
Sachsischen Gesellschaft der Wissenschaften, vol. 20, No. 2. Leipzig, 1893- 


Analysis: The Goniometric Functions. 

tion. Three points on a sphere, no two of which are diametrially opposite, 
determine uniquely a triangle in which each angle and each side lies 
between and n (see Fig. 70). Further investigation discloses that it 
is desirable to think of the sides and of the angles as unrestricted vari- 
ables, which can thus be greater than n or 2 n , or multiples of these 
values. One has to do then with sides that overlap and with angles 
which wind multiply around their vertices. It becomes necessary there- 
fore to adopt conventions concerning the signs of these quantities as 
well as the sense in which they are measured. It is due to Mobius, 
the great geometer of Leipzig, that the importance of the principle 
of signs was consistently developed, and the way 
opened for the general investigation of these quan- 
tities under unrestricted variation. The part of 
his work which is of particular significance here 
is the Entwicklung der Grundformeln der spharischen 
Trigonometric in grosstmoglicher Allgemeinheit 1 . 

This determination of the sign begins with the 
assumption of a definite sense of rotation about a 
point A on a sphere in which the angle shall be 
called positive (see Fig. 71). If this sense is settled for one point, 
it is for every other point, since the first point can be moved con- 
tinuously to that other. It is customary to select the counterclockwise 
rotation as positive, whereby we think of ourselves as looking at the 

sphere from the outside. Secondly, we must 
assign a sense of direction to each great circle 
on the sphere. We cannot be satisfied with an 
initial determination for one great circle and 
the continuous moving of it into coincidence 
with any second great circle, because this coin- 
cidence can be effected in two distinct ways. 
On this account, we shall assign a sense of 
direction separately to each great circle which 
we consider, and we shall look upon one and 
the same circle as, in a sense, two different configurations according 
as we have assigned to it the one or the other direction. - With this 
understanding, each directed great circle a can be uniquely related 
to a pole P , namely to that one of its two poles, in the elementary 
sense, from which its sense of direction would appear positive. Con- 
versely, every point on the sphere has a unique polar circle with a 
definite direction. With these considerations, the polarizing process, so 
important in trigonometry, is uniquely determined. 

Fig. 71. 

1 Berichte liber die Verhandlungen der Koniglich Sachsischen Gesellschaft der 
Wissenschaften, mathematisch-physikalische Klasse, vol. 12 (i860). Reprinted in 
Mfibius, F., Gesammelte Werke, vol. 2, p. 71. Leipzig, 1886. 

Spherical Trigonometry. 

If now three points A , B , C on the sphere are given, we must still 
make certain agreements, before a spherical triangle with these vertices 
is uniquely dtermined. In the first place, the direction of each great 
circle through A , B , C must be assigned, and we must know how many 
revolutions are necessary in order to bring a point from B to C, from 
C to A , and from A to B . The lengths a,b, c, determined in this way, 
which may be arbitrary real quantities, are called sides of the spherical 
triangle. Of course they are thought of as drawn on a sphere of radius 
one. The angles are then defined as follows: oc is that rotation, about A 
in positive sense, which would bring the direction CA into coincidence 
with the direction A B , to which arbitrary multiples of ^ 2 n may be 
added. The other angles are defined ana- 
logously. If we now examine an ordinary 
elementary triangle, as shown in Fig. 72, 
and determine the directions of the sides 
so that a , b, c are less than n , we find that 
the angles a, /?, y are, according to our 
new definiton, the exterior angles instead 
of the interior angles as in the usual 
consideration of the elementary triangle. 

It has been known for a long while that by replacing the customary 
angles of a spherical triangle by their supplements, in this manner, the 
formulas of spherical trigonometry turn out to be more symmetrical 
and perspicuous. The deeper reason for this appears from the following 
consideration. The polarizing process described above, by virtue of the 
conventions of Mobius, furnishes uniquely, for every given triangle, 
another triangle called the polar triangle of the first; and it is easy to 
see that, in view of our new definition, this polar triangle has for its 
sides and angles the angles and sides, respectively, of the original tri- 
angle. According to our agreements, then, every formula of spherical 
trigonometry must still hold if we interchange in it a, b, and c with <x , P , 
and v , respectively, so that there must always be this simple symmetry. 
If, on the other hand, the sides and angles are measured in the usual 
way, this symmetry is lost ; for the relation between triangle and polar 
triangle depends upon how one chooses the sides and angles in a given 
case, and upon how one resolves the ambiguity of the pole in the case 
of a non directed given circle. 

It is clear now that, of the six parts of a spherical triangle defined 
in this way, only three can be independent continuous variables, e.g. two 
sides and the included angle. The formulas of spherical trigonometry 
represent a number of relations between these parts or, to be more 
exact, of algebraic relations between their twelve sines and cosines, in 
which only three of these twelve magnitudes can be allowed to vary 
arbitrarily, while the other nine depend algebraically upon them. If 

Klein, Elementary Mathematics. 12 

Analysis: The Goniometric Functions. 

we go over to the sine and cosine, we can ignore the additive arbitrary 
multiples of 2 n . Let us now think of trigonometry as the aggregate of 
all possible such algebraic relations of this kind. Then we can state its 
problem, according to the modern manner of thinking, as follows. If 
we interpret the quantities 

as coordinates in a twelve dimensional space R 12 , then the totality of 
those of its points which correspond to actually possible spherical tri- 
angles a , . . . , y represents a three-dimensional algebraic configuration 
M 3 of this JR 12 , and the problem is to study this M 3 in the R IZ . In this 
manner spherical trigonometry is coordinated with general analytic 
geometry of hyperspace. 

Now this M 3 must have various simple symmetries. Thus the 
polarizing process showed that the interchange oia,b,c with <*, P,y, 
always yielded a spherical triangle. Translated into our new language, 
this states that when one interchanges x lt x 2 , x 3 , y lf y 2 , y 3 with # 4 , x 5 , 
x B>y*>y5>y6> respectively, any point of M 3 goes over into another 
point belonging to it. Further, corresponding to the division of space 
into eight octants by the planes of the three great circles, there exists 
for any triangle seven auxiliary triangles whose parts arise from those 
of the initial triangle through change of sign and the addition of n . This 
yields for every point of M 3 seven further points whose coordinates 
x lf . . ., XQ arise as a result of sign change. The totality of these sym- 
metries leads to a certain group of substitutions and sign changes of 
the coordinates of j?? 12 , which transforms M 3 into itself. 

The most important question now is that concerning the algebraic 
equations which are satisfied by the coordinates of M 3t and which 
constitute the totality of trigonometric formulas. Since cos 2 a + sin 2 oc 
= 1 , we have, to start with, the six quadratic relations 

(1) *! + :v! = i, (1 = 1, 2, ...,6), 

or, speaking geometrically, six cylindrical surfaces .F (2) of order two 
passing through M 3 . 

Six further formulas are supplied by the cosine theorem of spherical 
trigonometry, which in our notation, is 

cos a = cos & cose sin 6 sine cos a, 
from which one gets by polarization 

cos<x = cos/? cosy sm/Ssiny cosa . 

These equations, together with the four others which arise through 
cyclic permutation of a,b,c and ot, /?, y determine, all told, six cubic 
surfaces F^ passing through M 3 : 

(2) x l = x 2 x 3 - y 2 y 3 * 4 , * 2 = x 3 x l - y 3 y^x, , x 3 = 

(3) * 4 = *s*6 - 

Spherical Trigonometry. 

Finally, we can make use of the sine theorem, which can be expressed 
by the vanishing of the minors of the following matrix 

sin a, sin 6, sine 
sin a , sin/J, siny 
or, written at length, 

t A\ f\ 

These expressions represent three quadratic surface F* 9 of which only 
two, to be sure, are independent. Thus we have set up altogether 
fifteen equations for our M 3 in R 12 . 

Now, in general, 12 3 =9 equations do not, by any means, 
suffice to determine a three dimensional algebraic configuration in R 12 . 
Even in the ordinary geometry of R z , not every space curve can be 
represented as the complete intersection of two algebraic surfaces. The 
simplest example here is the space curve of order three which requires 
for its determination at least three equations. It is easy to see that, 
in our case also, the nine equations (1) and (2) do not determine M 3 . 
It is well known, namely, that the sine theorem can be derived from 
the cosine theorem only to within the sign, which one then determines, 
ordinarily, by geometric considerations. We should like to know then 
how many, and which, of the trigonometric equations really determine 
our M 3 completely. In this connection I should like to formulate four 
definite questions to which the literature thus far appears to give no 
precise answer. It could be a worth-while task to investigate them thor- 
oughly. That would probably not be especially difficult, after one had 
acquired a certain skill in handling the formulas of spherical trigono- 
metry. My questions are: 

1. What is the order of M 3 ? 

2. What are the equations of lowest degree by means of which M 3 
can be completely represented? 

3. What is the complete system of linearly independent equations 
which represent M 3 , i.e., of equations / a = 0, . . ., f n = such that the 
equation of every other surface passing through M 3 could be written 
in the forih m^^ + . . . + m n f n = 0, where m lf . . ., m n are integers? 
It is possible that more equations may be needed here than in 2. 

4. What algebraic identities (so called syzygies) exist between these 
formulas f lf . . ., / n ? 

One could gain familiarity with these things by consulting in- 
vestigations which have been made in exactly the same direction but 
in which the questions have been put somewhat differently. These 
appear in the Gottingen dissertation 1 , 1894, of Miss Chisholm (now 

1 Algebraisch-gruppentheoretische Untersuchungen zur sphdrischen Trigono- 
metrie, Gottingen, 1895- 


Analysis: The Goniometric Functions. 

Mrs. Young), who, by the way, was the first woman to pass the normal 
examination in Prussia for the doctor's degree. The most noteworthy 
of Miss Chisholm's various preliminary assumptions is her selection of 
the cotangents of the half angles and sides as independent coordinates. 
Since tan (a/2) and likewise, of course, ctn (a/2), is a fundamental 
function, in terms of which sin a and cos a can be uniquely expressed, 
it is possible to write all the trigonometric equations as algebraic relations 
between ctn (a/2) , . . . , ctn (y/2) . The spherical triangles constitute now 
a three-dimensional configuration M 3 in a six dimensional space R Q 
which has ctn (a/2), . . ., ctn (c/2), ctn (a/2), . . ., ctn (y/2), as coordi- 
nates. Miss Chisholm shows that this M 3 is of order eight and that it 
can be fully represented as the complete intersection of three surfaces 
of degree two (quadratic equations) of R 6 ] and she investigates also 
the questions which arise here, which are analogous to those stated 

In my lectures on the hypergeometric function 1 , I called the group 
of formulas of spherical trigonometry which I have discussed above, 
and which connect the sines and the cosines of the sides and angles, 
formulas of the first kind, in distinction from an essentially different 
group of formulas which I called formulas of the second kind. The latter 
are algebraic equations between the trigonometric functions of the half 
angles and sides. In studying them it will be best to select the twelve 

a . a a . a 

cos , smy , . . .; cos ~ 2 - > sm 2 

as coordinates in a new twelve space R\% , in which the spherical triangles 
again constitute a three-dimensional configuration M'% . It is here that 
those elegant formulas appear which, at the beginning of the last 
century, were published independently and almost simultaneously by 
Delambre (1807), Mollweide (1808) and finally Gauss 1809 [in the Theoria 
motus corporum coelestium, No. 54 2 ]. These are twelve formulas which 
arise by cyclic permutation in: 

8 -\- y b c .ft v b c 

sin^p cos- sm sin' 

2,2 2 -r- ^ 

. a -^ a ' . <x a 

sin cos sin sin- 

2 2 2 ^ 



oc ' a ' a - a 

cos-- cos cos- o sm--- 

2 2 2 & 

1 Winter semester 1893 1894. Elaborated by E. Ritter. Reprinted Leipzig, 

2 Reprinted in Werke, Leipzig, 1906, vol. 7, p. 67- 

Spherical Trigonometry. 

That which is essential and new in them, as opposed to the formulas 
of the first kind, is the double sign, with respect to which the following 
is true. For one and the same triangle, the same sign, either the upper 
or the lower, holds for all twelve formulas, and there are triangles of 
both sorts. The Mg of spherical triangles in the above defined R' 12 satis- 
fies, in other words, two entirely different systems of twelve cubic 
equations each, and divides therefore into two separate algebraic con- 
figurations M 3 , for which the one sign holds, and M 3 , for which the 
other holds. By virtue of this remarkable fact these formulas take on 
the greatest significance for the theory of spherical triangles. They are 
much more than mere transformations of the old equations which might 
at most serve to facilitate trigonometric calculation. To be sure, De- 
lambre and Mollweidc did consider these formulas only from this practical 
standpoint. It was Gauss who had the deeper insight, for he draws 
attention to the possibility of a change of sign "if one grasps in its 
greatest generality the idea of spherical triangle". It seems to me 
proper, therefore, that the formulas should bear Gauss's name, even if 
he did not have priority of publication. 

It was Study who first recognized the full range of this phenomenon, 
and who developed it in his memoir of 1893, which I mentioned on p. 1 75. 
His chief result can be stated most conveniently if we consider the six 
space R Q which has for coordinates the quantities # , & , c , ft , /? , y them- 
selves, thought of as unrestricted variables. I call them transcendental 
parts of the triangle in destinction from the algebraic parts cos #,..., 
or cos (a/2) , . . . , because the former arc transcendental functions, while 
the latter are algebraic functions of the ordinary space coordinates of 
the vertices of the triangle. In this JR 6 , the aggregate of all spherical 
triangles appears as the transcendental configuration M ( > whose image 
in R\2 is the algebraic M' 3 considered above. Since however the latter 
split into two parts and the mapping functions cos (a/2) , . . . are single 
valued continuous functions of the transcendental coordinates, the trans- 
cendental M^ must split into at least two separated parts. Study's 
theorem is as follows : The transcendental configuration M^ of the quan- 
tities a,b,.c, <*, P,y, belonging to a spherical triangle of the most general 
sort, divides into two separate parts corresponding to the double sign in 
the Gaussian formulas, and each of these parts is a connected continuum. 
The essential thing here is the exclusion of any farther division. It 
would not be possible, by farther manipulation of the trigonometric 
formulas, to bring about similar and equally significant groupings of 
spherical triangles. The triangles of the first of these parts, that corre- 
sponding to the upper sign in the Gaussian formulas, are called proper 
triangles, those of the other, improper, and we may state Study's 
theorem briefly as follows : The totality of all spherical triangles resolves 
itself into a continuum of proper and one of improper triangles. You 

Analysis: The Goniometric Functions. 

will find further details, and a proof of this theorem, in Weber- 
Wellstein 1 . I am attempting here only to state the results clearly. 
I must now say something further concerning the difference between 
the two sorts of triangles. If a spherical triangle is given, i.e., an ad- 
missible set of values of#,&,c,<x,/?,y, whose cosines and sines satisfy 
the formulas of the first sort, and which therefore represents a point 
of M^, how can we decide whether the triangle is proper or improper? 
In order to answer this question we first find the smallest positive 
residues a ,b ,c , <x , /3 , y of the given numbers, with respect to the 
modulus 2n\ 

# = a (mod 2n) , . . . , <X Q = oc (mod 2n) , . . . 
Q^a Q <27i, . . . , 0^<x < 2rc, . . . 

Their sines and cosines coincide with those of a , . . . , a , . . . so that 
they also represent a triangle which we shall call the reduced, or the 
Moebius, triangle corresponding to the given one, since Moebius himself 
did not consider the parts as varying beyond 2 n . Then we can deter- 
mine, by means of a table, whether the Moebius triangle is proper or 
improper. You will find this, in a form somewhat less clear, in Weber- 
Wellstein (p. 352, 379, 380), as well as figures (p. 348, 349) of the types 
of proper and improper triangles. As is usual, I shall call an angle 
reentrant when it lies between n and 2 n and I shall, for the sake of 
brevity, apply this term also to the sides of the spherical triangle. Then 
there are, altogether, four typical cases of each sort. 

I. Proper Moebius triangles'. 

1. sides reentrant; angles reentrant. 

2. 1 side reentrant; 2 adjacent angles reentrant. 

3. 2 sides reentrant; 1 included angle reentrant. 

4. 3 sides reentrant; 3 angles reentrant. 

II. Improper Moebius triangles'. 

1. sides reentrant; 3 angles reentrant. 

2. 1 side reentrant; 1 opposite angle reentrant. 

3. 2 sides reentrant; 2 opposite angles reentrant. 

4. 3 sides reentrant; angles reentrant. 

There are no cases other than these, so that this table enables us actually 
to determine the character of a Moebius triangle. 

The transition to the general triangle a , . . . , oc, t . . . from the cor- 
responding reduced triangle is made, after what was said above, by 
means of the formulas: 

a = + n^ 2n , b = b Q + n 2 *2n , c = C Q + n% 2n , 

* = #o + v \ ' 2n > P = A) + V 2 ' 2n > y = 7o + V 3 ' 2n 
1 Vol. 2, second edition (1907), p. 385 ( 47). 

Spherical Trigonometry. 

We may then make use of the following theorem The character of the 
general triangle is the same as or the reverse of that of the reduced triangle 
according as the sum of the six integers n + n 2 + n 3 + v l -f- ^ 2 + V 3 ^ s 
even or odd. Thus the character of every triangle as proper or improper 
can be determined. 

I shall conclude this chapter with a few remarks about the area 
of spherical triangles. Nothing is said about this in Study or in Weber- 
Wellstein. It does come up for consideration in my Alteren funktionen- 
theoretischen Untersuchungen uber Kreisbogenfireiecke* . Up to this point 
we have considered the triangle merely as an aggregate of three angles 
and three sides which satisfy the sine and consine laws. In my in- 
vestigations I was concerned with a definite area bounded by these 
sides, in a certain sense with a membrane stretched between these 
sides and involving appropriate angles. 

Of course we can now no longer think of <x , ft , y as the exterior 
angles of the triangle, as we did before for reasons of symmetry. We 
shall talk, rather, of those angles which the membrane itself forms at 
the vertices, and I shall call them interior angles of the triangle. I shall 
denote them, as is my habit, by ATT, //TT, vn (see Fig. 73). These angles 
can also be thought of as unrestricted positive variables, since the 
membrane might wind about the vertices. 
In accordance with this, I shall denote the 
absolute lengths of the sides by In , mn , n n , 
which are also unrestricted positive variab- 
les. But it will be no longer possible for 
the sides and the angles to "overlap" in- 
dependently of one another, i.e., to contain 
arbitrary multiples of 2 n , as they could 
before, for the fact that a singly-connected F te- 73. 

membrane should exist with these sides 

and angles finds its expression in certain relations between the numbers 
of these overlappings. In my memoir Uber die Nullstellen der hyper- 
geometrischen Reihe 1 I called these supplementary relations of spherical 
trigonometry. If we denote by E (x) the largest positive integer which x 
exceeds, [E (x) < x] , these relations are 

* Earlier function-theoretic investigations of spherical triangles. 
1 Mathematische Annalen, vol. 37 (1888). [Reprinted in Klein, F., Gesammelte 
Mathematische Abhandlungen, vol. 2 (1921), p. 550. 

Analysis: The Goniometric Functions. 

and since E (1/2) , for example, gives the multiple of 2 n which is contained 
in the side / n t these relations determine precisely the desired "overlap* ' 
numbers of the sides ln,mn t nn when one knows the angles AJT, ^n y vn 
together with their overlap numbers. It is easy to see, in particular, 
that of the three numbers A // v , l + p v, I p + v, one 
at most can be positive. Consequently only one of the three arguments 
on the right sides can exceed unity, and since E (x) = f or x ^ \ , it 
is possible for only one of the overlap numbers to be different from zero. 
In other words only one side, at most, of a triangular membrane can 
overlap (be greater than 2) and that side must be opposite the largest 

For the proof of these supplementary relations I refer you to my 
mimeographed lectures Uber die hypergeometrische Funktion 1 (p. 384), 
although the edition is long since exhausted. There, as well as in my 
memoir in volume 37 of the Mathematische Annalen, the initial assump- 
tions were somewhat broader than the present ones, in that spherical 

triangles were considered which are bounded 
by arbitrary circles on the sphere, not neces- 
sarily by great circles. I shall sketch briefly 
the train of thought of the proof. We start 
with an elementary triangle, in which a 
membrane can certainly be stretched, and 
obtain from it step by step the most general 
admissible triangular membrane by repe- 
, atedly attaching circular membranes, either 

at the sides, or, with branchpoints, at the 
Fig . 74 . vertices. Fig. 74 shows, as an example, (in 

stereographic projection) a triangle ABC 

which arises from an elementary triangle by attaching the hemi- 
sphere which is bounded by the great circle AB, whereby the side AB 
overlaps as well as the angle C. It is clear that the supplementary 
relations continue to hold here, and one sees in the same way that 
they retain their validity for the most general triangular membrane 
which can be built up by this process. 

We must now inquire how these triangles, which satisfy the supple- 
mentary relations, fit the general theory which we have discussed 
already. They are obviously only special cases, (because the overlap 
numbers of the sides and angles are, in general, entirely arbitrary) 
special cases which are characterized by the possibility of framing a 
stretched membrane in a triangle. At first one can really be puzzled 
here, for we have seen that the totality of all proper triangles (some of 
which do not need to satisfy the supplementary relations) constitutes 

1 These lectures were referred to on p. 180. 

Spherical Trigonometry. 


Fig. 76. 

a continuum, and that any one of them could be derived, therefore, 
from an elementary triangle by a continuous deformation. One would 
think, naturally, that it would be impossible, during this deformation 
to lose the membrane which was stretched in the initial elementary 
triangle. The explanation of this difficulty appears if we extend Moebius' 
principle of sign-change to areas, by agreeing that an area is to be called 
positive or negative according as its boundary is traversed in the positive 
(counter clockwise) or negative sense. Accordingly, when a curve which 
crosses itself bounds several partial areas, the entire area is the algebraic 
sum of the several parts, each of these determined, as to sign, by the 
sense in which its boundary is traversed. In Fig. 75 this would be the 
difference, in Fig. 76, the sum of the parts which are distinguished by 
different shading. These agreements are, of 
course, merely the geometric expression of that 
which the analytic definition itself supplies. 

If we apply this, in particiilar, to triangles 
formed by circular 
arcs, it turns out, in 
fact, that with every 
proper triangle we can 
associate an area on 
the sphere such that, when one circuit of the triangle is made, 
different parts of this area are combined with different signs 
because the boundaries of these parts are traversed in different senses. 
Those triangles for which the supplementary 
relations hold are special, then, in that their 
areas consist of a single piece of membrane 
bounded by a positive circuit. It is this pro- 
perty which gives them their great significance 
for the function-theoretic purposes to which 
I put them in my earlier studies. 

I will now illustrate this situation by means 
of an example. Let us consider the triangle 
ABC in t stereographic projection (Fig. 77) 
where, of the points of intersection A , A' of 
the great circles BA, CA, A is the one more 
remote from the arc B C . If we now transfer the 
general definition of the exterior angles (p. 177) to their supplements, the 
interior angles, we find that [An and vn measure the rotation of J5Cinto 
BA and of CA into CB, respectively, and are, therefore, positive in our 
case. Similarly kn measures the rotation of AB into AC and is there- 
fore negative. Put A = - A', K > 0. Then the triangle A'BC is ob- 
viously an elementary triangle with angles A'rc, ^n t vn , all of which 
are positive. If we now make a circuit about the triangle ABC , the 

Fig 77- 

186 Analysis: The Goniometric Functions. 

boundary of the elementary triangle A'BC will be traversed in the 
positive sense but that of the spherical sector A A' in the negative, and 
the area of the triangle ABC , in the Moebius sense, will be the difference 
of these two areas. This breaking up of the triangular membrane into 
a positive and a negative part can be visualized, perhaps, by supposing 
the membrane twisted at A' so that the rear or negative side of the 
sector is brought to the front. It is not hard to construct more difficult 
examples after this pattern. 

I shall now show, by means of this same example, that with this 
general definition of area, the formulas for the area of elementary tri- 
angles still remain valid. As you know, the area of a spherical triangle 
with angles kn t [JLTI, vn t on a sphere with radius one, is given by the so- 
called spherical excess (A -f // + v 1) n where A, /j, v > 0. Let us 
now see that this formula holds also for the above triangle ABC. It 
is clear that the area of the elementary triangle A 'B C is (A' -f // + v \ } n. 
From this we must subtract the area of the sector A A' whose angle 
is A'TT. But this is 2 I'M, because the area of a sector is proportional to 
its angle; and it becomes 4n when the angle is 2 n (the entire sphere). 
We get then, as the area of ABC, 

(X + p + v \}n-2Vn = (V + /LL + V \)n= (i + p + v \)n. 

It is probable, if we had a general proper triangle with arbitrary sides 
and angles, and if we should try to fit into it a multi-parted membrane 
and determine its area (which, according to the sign rule, would be 
the algebraic sum of the parts), that the result would show the general 
validity of the formula (A -}- /j + i> \)n, where, of course, Arc, . . . 
are the real angles of the membrane, and not, as before, the exterior 
angles. The investigation suggested here has not been carried out, 
however. It would certainly not offer great difficulties, and I should 
be glad if it were undertaken. At the same time, it would be important 
to determine, from the present standpoint, the role of the improper 

With this I shall leave the subject of trigonometry and go over to 
the second important application of goniometric functions, one which 
also falls within the field of the schools. 

B. Theory of small oscillations, especially those of the 


I shall recall briefly the deduction of the law of the pendulum as we 
are in the habit of giving it at the university, by means of infinitesimal 
calculus. A pendulum (see Fig. 78) of mass m hangs by a thread of 
length /, its angle of deflection from the normal being <p. Since the 
force of gravity acts vertically downwards, it follows from the funda- 

Small Oscillations. jgj 

mental laws of mechanics that the motion of the pendulum is deter- 
mined by the equation 

(1) . 

For small amplitudes we may replace sin <p by <p without serious error. 
This gives for so called infinitely small oscillation of the pendulum 

The general integral of this differential equa- 
tion is given, as you know, by goniometric func- 
tions, which are important here, as I said before. 
precisely by reason of their differential properties 
The general integral is 


where A , B are arbitrary constants. If we introduce appropriate new 
constants C,/ , we find 

(3) p = C 

where C is called the amplitude and / the phase of the oscillation. 
From this we get, for the duration of a complete oscillation, T 2n]/l/g. 

Now these are very simple and clear considerations, and if we went 
more fully into the subject they could of course be given graphical 
form. But how different they appear from the so called elementary 
treatment of the pendulum law which is widely used in school instruction. 
In this, one endeavors, at all costs, to avoid a consistent use of infinitesi- 
mal calculus, although it is precisely here that the essential nature of 
the problem demands emphatically the application of infinitesimal 
methods. Thus one uses methods contrived ad hoc, which involve 
infinitesimal notions without calling them by their right name. Such 
a plan is, of course, extremely complicated, if it is to be at all exact. 
Consequently it is often presented in a manner so incomplete that it 
cannot be thought of, for a moment, as a proof of the pendulum law. 
Then we have the curious phenomenon that one and the same teacher, 
during one hour, the one devoted to mathematics, makes the very 
highest demands as to the logical exactness of all conclusions. In his 
judgment, still steeped in the traditions of the eighteenth century, his 
demands are not satisfied by the infinitesimal calculus. In the next 
hour, however, that devoted to physics, he accepts the most questionable 
conclusions and makes the most daring application of infinitesimals. 

To make this clearer, let me give, briefly, the train of thought of 
such an elementary deduction of the pendulum law, one which is actually 
found in text books and used in instruction. One begins with a canonical 

-jgg Analysis: The Goniometric Functions. 

pendulum, i.e. a pendulum in space whose point moves with uniform 
velocity v in a circle about the vertical, as axis, so that the suspending 
thread describes a circular cone (see Fig. 79). This is the motion which 
is called in mechanics regular precession. The possibility of such motion 
is, of course, assumed in the schools as a datum of experience and the 
question is asked merely concerning the relation which obtains between 
the velocity v and the constant deflection of the pendulum, cp = oc (angular 
opening of the cone which is described by the thread). 

One notices, first, that the point of the pendulum describes a circle 
of radius r = I sin oc , for which one may write r = I oc when oc is 

sufficiently small. Then one talks of cen- 
trifugal force and reasons that the point, 
with mass m, revolving with velocity v , 
must exert the centrifugal force 

z; 2 v 2 

m = m -, 
r I - a 

In order to maintain the motion there 
must be an equal centripetal force directed 
toward the center of the circular path. 
Fig. 79. This is found by resolving the force of 

gravity into two components, one directed 

along the thread of the pendulum, the other, the desired force, acting 
in the plane of the circular path and directed toward its center, having 
the magnitude m g tan oc (see Fig. 79). This can be replaced by mg - oc 
when oc is sufficiently small. We obtain, then, the desired relation in 
the form 

m -- = mg oc , or v = oc 1/g / . 
lot, ' 

The time of oscillation T of the pendulum, that is, the time in which 
the entire circumference of the circle 2nr = 2nloc is traversed, is then 

= 2 


In other words, when the angle of oscillation oc is sufficiently small, 
the canonical pendulum performs a regular precession in this time, 
which is independent of oc. 

To criticize briefly this part of the deduction, we might admit the 
validity of replacing sin oc and tan oc by oc itself, which we did ourselves 
in our exact deduction (p. 187); for this permits the transition from 
"finite" to "infinitely small" oscillations. On the other hand, we must 
call attention to the fact that the formula used above for centrifugal 
force can be deduced in "elementary" fashion only by neglecting all 
sorts of small quantities; and the exact justification for this is founded 
precisely on differential calculus. The very definition of centrifugal 

Small Oscillations. 


force, for example, requires in fact the notion of the second differential 
coefficient, so that the elementary deduction must also smuggle this in. 
And since in doing this, one is unable to say clearly and precisely what 
one is talking about, there arise the greatest obstacles to understanding, 
which are not present at all when the differential calculus is used. I do 
not need to go into detail here because I can refer you to some very 
readable articles on school programs, by the deceased realgymnasium 
director H. Seeger 1 , in Gustrow and to a very interesting study by 
H. E. Timerding: Die Mathematik in den 
physikalischen Lehrbuchern 2 . In Seeger you 
will find, among other things, an exhaustive 
criticism of the deductions of the formula 
for centrifugal force, in a manner corre- 
sponding to our standpoint. In Timerding 
there are extensive studies of the mathe- 
matical methods which are traditionally used 
in the teaching of physics*. Let me now 
continue with the discussion of pendulum 

The considerations set forth above show the possibility of uniform 
motion in a circle. If we now set up an x y coordinate system (see 
Fig. 80) in the plane of this circle (i.e., in view of our approximation, 
the tangent plane to the sphere), this motion will, in the language of 
analytic mechanics, be given by the equations 


*-/* cos J/|-(/-g 

y = / * sin J/| (t - t Q ) 

But we wish the plane oscillations of the pendulum; that is, the 
point of the pendulum in our x y plane is to move on a straight line, 
the x axis. The equations of its motion must be 


= 0, 

1 Vber die Stellung des hiesigen Realgymnasiums zu einem Beschlusse der letzten 
Berliner Schulkonferenz (Gustrow, 1891, Schulprogramm No. 649). Vber die Stellung 
.des hiesigen Realgymnasiums zu dem Erlass des preussischen Unterrichtsministeriums 
von 1892 (1893, No. 653)- Bemerhungen uber Abgrenzung und Verwertung des 
Unterrichts in den Elementen der Infinitesimalrechnung (1894, No. 658). 

2 Bd. Ill, Heft 2 der "Abhandlungen des deutschen Unterausschusses der 
Intern ationalen mathematischen Unterrichtskom mission' 1 . Leipzig u. Berlin 1910. 

* See also Report on the Correlation of Mathematics and Science Teaching 
by a joint committee of the British Mathematical Association and the Science 
Masters Association 1908. Reprinted 1917. Bell and Sons, London. 

Analysis: The Goniometric Functions. 

in order that the correct equation (3) shall result when <p = x/l. Thus 
we must pass from equations (4) to (5) without, however, making use 
of the dynamical differential equations. This is made possible by setting 
up the principle of superposition of small oscillations, according to which 
the motion x + #i> y + y\ is possible when the motions x, y and x lf y l 
are given. We may combine, namely, the counterclockwise pendulum 
motion (4) with the clockwise motion 

Xl = I . oc cos j/^ (t g , y l = Z (x sin j/ 1- (t t ) . 

Then, if we put a = C/2, the motion x + x lt y + y l is precisely the 
oscillating motion (5) which was desired. 

In criticizing what precedes, we inquire, above all, how the principle 
of superposition is to be established, or at least made plausible, without 
the differential calculus. With these elementary presentations there 
remains always the doubt as to whether or not our neglecting of suc- 
cessive small quantities may not finally accumulate to a noticeable error, 
even if each is permissible singly. But I do not need to carry this out 
in detail, for these questions are so thoroughly elementary that each 
of you can think them through when you feel so inclined. Let me, in 
conclusion, state with emphasis that we are concerned in this whole 
discussion with a central point in the problem of instruction. First, 
the need for considering the infinitesimal calculus is evident. Moreover, 
it is clear that we need also a general introduction of the goniometric 
functions, independently of the geometry of the triangle, as a preparation 
for such general applications. 

I come now to the last of the applications of the goniometric functions 
which I shall mention. 

C. Representation of periodic functions by means of series 
of goniometric functions (trigonometric series) 

As you know, there is frequent occasion in astronomy, in mathe- 
matical physics, etc., to consider periodic functions, and employ them 
in calculation. The method indicated in the title is the most important 

and the one most frequently 
used. For convenience we 
shall suppose the unit so 
chosen that the given pe- 
riodic function y / (x) 

Fig. si. has the period 2n (see 

Fig. 81). The question then 

arises as to whether or not we can approximate to this function by 
means of a sum of cosines and sines of integral multiples of x, from 
the first, to the second,..., in general to the w-th, each, with a 

Trigonometric Series. 

properly chosen constant factor. In other words, can one replace f(x), 
to within a sufficiently small error, by an expression of the form 

S n (x) = ~ + a 1 cos x + a 2 cos 2x + + a n cosnx 

+ b l sinx + b 2 s\n2x + + b n sinnx. 

The factor \ is added to the constant term to enable us to give a general 
expression for the coefficients. 

First I must again complain about the presentation in the text books, 
this time the texts in differential and integral calculus. Instead of 
putting into the foreground the elementary problem which I have 
outlined above, they often seem to think that the only problem which 
is of any interest at all is the theoretical question, connected with the 
one we have raised, whether / (x) can be exactly represented by an 
infinite series. A notable exception to this is Runge in his Theorie und 
Praxis der Reihen 1 . As a matter of fact, that theoretical question is, 
in itself, thoroughly uninteresting for practical purposes, since we are 
concerned in practice with a finite number of terms, and not too many 
at that. Moreover it does not even permit a conclusion a posteriori as 
to the practical usableness of the series. One may by no means conclude 
from the convergence of a series that its first few terms afford even a 
fair approximation to the sum. Conversely, the first few terms of a 
divergent series may be useful, under certain conditions, in representing 
a function. I am emphasizing these things because a person who knows 
only the usual presentation and who wishes then to use finite trigono- 
metric series in, say, the physical laboratory, is apt to be deceived 
and to reach conclusions that are unsatisfactory. 

The customary neglect of finite trigonometric sums seems still more 
remarkable when one recalls that they have long been completely treated. 
The astronomer Bessel gave the authoritative treatment in 1815- You 
will find details concerning the history and literature of these questions 
in the encyclopedia reference by Burkhardt on trigonometrische Inter- 
polation (Enzyklopadie II A 9, p. 642 et seq.). Moreover, the formulas 
that concern us here are essentially the same as those that arise in the 
usual convergence proofs. It is only that the thoughts which we shall 
attach to them have another shade of meaning and are designed to 
adapt the material more for practical use. 

I turn now to a detailed consideration of our problem, and I shall 
inquire first as to the most appropriate determination of the coefficients 
a, 6, ... for a given number n of terms. Bessel developed an idea here 
which involves the method of least squares. The error that is made 
when, for a particular x, we replace / (x) by the sum S n (x) of the first 

1 Sammlung Schubert No. 32, Leipzig, 1904. See also Byerly, W. E., Fourier's 
Series and Spherical Harmonics. 


Analysis: The Goniometric Functions. 

2 n + 1 terms of the trigonometric series, is / (x) S n (x) , and a measure 
of the closeness of representation throughout the interval ^ x ^ 2 n 
(the period of / (x)) will be the sum of the squares of all the errors, that 
is, the integral 

The most appropriate approximation to / (x) will therefore be supplied 
by that sum S n (x) for which this integral / has a minimum. It was 
from this condition that Bessel determined the 2n + \ coefficients a , 
a lt . . <, a n ,b lt . . ., b n . Since we are to consider / as a function of the 
2n + 1 quantities , . . ., b n , we have, as necessary conditions for a 


Since / is an essentially positive quadratic function of a Q , . . ., b n , it is 
easy to see that the values of the variables determined by these 2n + 1 
equations really yield a minimum. 

If we differentiate under the sign of integration, the equations (2) 
take the form 



/ [/(*) S n (* 

= O l ... l f '" r [/W-S ll (^)]si 

Now the integrals of the products of S n (x) by a cosine or a sine can 
be much simplified. We have, namely, for v = 0, 1 , . . . , n, 

I S n (x)cosvxdx = ^ cosvxdx + a* co$xcosvxdx+"*+a n ^cosnxcosvxdx 

Jo 2 JO JO JO 

/2jr r2n 

+ 6J smxcosvxdx+'"+b n l sinnxcosvxdx. 
Jo Jo 

According to known elementary integral properties of the goniometric 
functions, all the terms on the right vanish, with the exception of the 
cosine term with index v, which takes the value a v *n, so that 

S n (x)cosvxdx = i 

(v = 0, 1, . . . , n) . 

Trigonometric series. 

This result holds also for v = 0, by virtue of our having given to a 
the factor . Similarly, we have also 

/ S n (x) sinvxdx = b v n , (v = 1, . . . , n) . 


From these simple relations, it follows that each of the equations (2') 
contains only one of the 2^ + 1 unknowns. We can therefore write 
down their solutions immediately in the form 


j r2n 

a v= f(x)cosvxdx, (v = 0, 1, . . . , n), 

\ r 2n 

b v = / f(x) sinvxdx, (v = 1, . . . , n) . 

ft J o 

If we make use of these values of the coefficients in S n (x) , as we 
shall from now on, / actually becomes a minimum, and its value is 
found to be 



It is important to notice that the values of the coefficients a , b 
which result from our initially assumed form of S n (x) are independent 
of the special number n, and that, furthermore, the coefficient belonging 
to a term cosrx or sin vx has precisely the same value, whether one 
uses this term alone or together with any of the others, in approximating 
to / (x) according to the same principle. If we attempt, namely, to 
make the best possible approximation to / (x) means of a single cosine 
term a v cosvx, that is, so that 


[f(x) a v cosvx]*dx = Minimum 

we find for a v the same value that was deduced above. This fact makes 
this method of approximation especially convenient in practice. If, 
for example, one has been led to represent a function by a single multiple 
of sin x, because its behaviour resembled the sine, and finds that the 
approximation is not close enough, one can add on more terms, always 
according to the principle of least squares, without having to alter the 
term already found. 

I must now show how the sums S n (x) , determined in this way, 
actually tend toward the function / (x) . For such an inquiry it seems 
to me desirable to proceed, in a sense, experimentally, after the method 
of natural scientists, namely by first drawing for a few concrete cases 
the approximating curves S n (x) . This gives a vivid picture of what 
happens, and, even for persons without special mathematical gift, it 
will awaken interest, and will show the need of mathematical explanation. 

Klein, Elementary Mathematics. 1 3 

194 Analysis: The Goniometric Functions. 

In a former course of lectures (Winter semester 1903-1904) when 
I discussed these things in detail, my assistant, Schimmack, made such 
drawings, some of which I shall show you in the original and on the 

1. We get simple and instructive examples of the desired kind if 
we take curves made up of straight line segments. For example, consider 
the curve y = / (x) as coinciding with y = x, from x = to x = n\2\ 
with y = n x , from x = nj2 to x = 3 nj2 \ with y = x 2 n from 
x = 3 nj2 to x = 2 n ; and as periodically repeating itself beyond the 
interval considered (0,2^). If we calculate the coefficients, we find 
all the coefficients a v are zero, since / (x) is an odd function, and there 
remain only the sine terms. The desired series has the form 

c/^\ _ 4 / sin * sin 3* , sins* \ 

. ^w-^ni 31- + 5f--+ -; 

In Fig. 82 the course of the first and second partial sums is sketched. 
The partial sums approach the given curve y = f (x) more and more 

Fig. 82. 

closely in that the number of their intersections with it increase continually 
It should be noticed especially that the approximating curves crowd 
more and more into the corners of the curve at nj2 , 3 ^/2 , . . . , although 
they themselves, as analytic functions, can have no corners. 

2. Let / (x) be defined as x from x = to x = n , and as x 2 n 
from x = ntox = 2n, with a gap at x = n . The curve consists, then, 
of parallel straight line segments through the points x = Q,2n,4tt, ... 
of the x axis. If at the points of discontinuity we insert vertical lines 
joining the ends of the discontinuous segments, the function will be 
represented by an unbroken line (see Fig. 83). It looks like the m strokes 
which you all practiced when you were laming to write. Again the 
function is odd, so that the cosine terms drop out, and the series becomes 

Fig. 83 represents the sums of the first two, three, and four terms. 
It is especially interesting here, also, to notice how they try to imitate 

Trigonometric series. 


the discontinuities of / (x), e.g., by going through zero at x = n with 
ever increasing steepness. 

3. As a last example (see Fig. 84) I shall take a curve which is equal 
to Tt/2 between and n/2, 
equal to between a/2 and 
371/2, and finally equal to 
n/2 between 3 ^/2 and 
2^, and which continues 
periodically beyond that. 
If we again insert verti- 
cal segments at the places 
of discontinuity we get a 
hookshaped curve. Here 
also only the sine coeffi- 
cients are different from 
zero, since we have an 
odd function, and the series 
becomes Fig. 83. 

S(x) = sinx + 2 

sin 2x sin 3 A; 

+ /} 

sin 6 AT 

sin ix 

The law of the coefficients is not so simple here as it was before and 
hence the successive approximating curves (Fig. 84 shows the third, 
,. > _ x _^ fifth and sixth) are not so comparable gra- 

phically as they were in the preceding cases. 
We turn now to the question as to how 
large the error is, in general, when we replace 

Fig. 84. 

f (x) , at a definite place, by the sum S n (x) . Up to this point we have 
been concerned only with the integral of this error, taken for the entire 
interval. Let us consider the integrals (3) (p. 193) f r th e coefficients 
a vt b v and replace the variable of integration by I, to distinguish it 


Analysis: The Goniometric Functions. 

from x, which we use to denote a definite point. Then we can write 
our finite sum (1) as 

1 /* 2 
n (x) = / 


or, if we combine summands which are in the same column, we have 

(*-)+cos2(*-{) + + cos(*-|)]. 

The series in the parenthesis can be summed easily, perhaps most con- 
veniently by using the complex exponential function. I cannot go into 
the details here, but we get, if we also use the fact that the periodicity 
of the integrand enables us to integrate from n to -\-n\ 

. 2n + \ ... . 

-*) . 

To enable us to judge as to the value of this integral, let us first draw 
the curves 

, l 1 


for the interval x n^S^lx + n of the axis. They obviously 
have branches resembling a. hyperbola (see Fig. 85), and between these 
branches the curve 

2n + j ft _ 

oscillates back and forth with increasing frequency as n gets larger. 
For = x it has the value r\ = (2n + l)/(2^) which increases with n. 

/ + JT 
7? '^1 

will represent simply the area lying between the r\ curve and the f axis 
(shaded in the figure). Now anyone who has moderate feeling for con- 
tinuity will see at once that if n increases sufficiently the* oscillation 
areas to the right, as well as those to the left, being alternately positive 
and negative, will compensate each other and that only the area of 
the long narrow central arch will remain. But it is easy to see that 
with increasing n this approaches the value / (x) == 1 , as it should. 
And, in general, things turn out in this same way, provided / (x) does 
not oscillate too strongly at x = f . 

It is just such considerations, developed for more precise use, which 
form the basis for Dirichlet's proof of convergence of the infinite trigono- 
metric series. 

Trigonometric series. 


This proof was published 1 for the first time by Dirichlet in 1829 
in volume 4 of Crelle's Journal. Later (1837) he gave a more popular 
presentation 2 in the Repertorium der Physik by Dove and Moser. The 
proof is given nowadays in most textbooks*, and I do not need to 
dwell upon it here. But I must mention certain sufficient conditions 
which the function / (x) must satisfy if it is to be represented by an 
infinite trigonometric series. Again think of / (x) as given in the interval 

Fig. 85- 

x ^ 2 n and as periodically continued beyond. Dirichlet makes, 
then, the following two assumptions which are called today simply 
Dirichlet' s conditions : 

a) The given function / (x) is segmentally continuous, i.e., it has in 
the interval (0, 2 n) only a finite number of discontinuities, and is other- 
wise continuous up to the points where it jumps. 

b) The given function / (x) is segmentally monotone, i.e., one can 
divide the interval (0,2^) into a finite number of sub-intervals, in 
every one of which / (x) either does not increase or does not decrease. 
In other words, / (x) has only a finite number of maxima and minima. 
(This would exclude, for example, such a function as sin \jx, for which 
x = o is a limit point of extrema.) 

Dirichlet shows that, under these conditions, the infinite series re- 
presents th*e function / (x) exactly for all values of x for which / (x) is 

continuous. That is 

limS, (*)=/(*). 


Moreover Dirichlet proves that, at a point of discontinuity, the series 
converges also, but to a value which is the arithmetic mean of the two 

1 Reprinted in Dirichlet, Werke, vol. 1, p. 117, Berlin, 1889. 

? Vber die Darstellung ganz willkurlicher Funktionen durch Sinus- und 
Kosinusreihen. Reprinted, Werke, vol. 1, p. 133 160, and Ostwalds Klassiker 
No. 116, Leipzig, 1900. 

* See Byerly, Fourier's Series and Spherical Harmonics. 


Analysis: The Goniometric Functions. 

values which / (x) approaches when x approaches the discontinuity Irom 
the one side or the other. This fact is usually expressed in the form 

Fig. 86 exhibits such discontinuities and the corresponding mean values. 
These conditions of Dirichlet are sufficient, but by no means ne- 
cessary, in order that / (x) may be represented by the series 5 (x) . On 
the other hand, mere continuity of / (x) is not sufficient. In fact it is 
possible to give examples of continuous functions where oscillations 
cluster so strongly that the series S (x) diverges. 

After these theoretical matters I shall now return to the practical 
side of trigonometric series. For a detailed treatment of the questions 
that arise here I refer you to the book by Runge which I mentioned 

before (see p. 191). You will find 
there a full treatment of the 
question as to the numerical cal- 
culation of the coefficients in the 
series, i.e., the question as to 
how, when a function is given, 
one can rapidly evaluate the 
integrals for a V9 b v in the most 




Fig. 86. 

suitable way. 

Special mechanical devices called harmonic analyzers have been 
constructed for calculating these coefficients. This name has reference 
to the relation which the development of a function / (x) into a trigono- 
metric series has to acoustics. Such a development corresponds to the 
separation of a given tone y = f (x) (where x is the time and y the 
amplitude of the tone vibration) into "pure tones", that is, into pure 
cosine and sine vibrations. In our collection we have an analyzer by 
Coradi in Zurich, by means of which one can determine the coefficients 
of six cosine and sine terms (v = 1 , 2, . . . , 6), i.e. twelve coefficients 
in all. The coefficient <z /2 must be separately determined by a plani- 
meter. Michelson and Stratton have made an apparatus with which 
160 coefficients (v = 1 , 2, . . . , 80) can be determined. It is described 
in Runge's book. Conversely, this apparatus can also sum a given 
trigonometric series of 160 terms, i.e. calculate the function from the 
given coefficients a v , b v . This problem also , of course, is of the greatest 
practical importance. 

The apparatus of Michelson and Stratton called attention anew to 
a very interesting phenomenon, one which had been noticed earlier 1 but 

1 According to Enzyklopadie vol. 2, 12 (Trigonometrische Reihen und Integrate), 
p. 1048, H. Wilbraham was already familiar with the phenomenon under discussion 
here and had treated it with a view to calculation. 

Trigonometric series. 


Fig. 87. 

which, with the passage of decades, had, curiously enough, been forgotten. 
In 1899 Gibbs again discussed it in Nature 1 , whence it is called Gibb's 
phenomenon. Let me say a few words about it. The theorem of Birichlet 
gives as the value of the infinite series, for a fixed value x, the expression 
[/ (x + 0) + / (x 0)] . In the second example discussed above (to have 
a concrete case in mind) the series gives the values at the isolated 
points n, 3 n, . . . of the function pictured in Fig. 87. 

Now the way in which we explained the matter of trigonometric 
approximation was different from the Dirichlet procedure, where x is 
kept fixed while n becomes infinite. We 
thought of n as fixed, considered S n (x) with 
variable x , and drew the successive approx- 
imating curves 5 X (x) , S 2 (x) , S 3 (x) , . . . We 
may now inquire, what happens to these 
curves when n becomes infinite; or, to put 
it arithmetically, what is the limit of S n (x) 
when n becomes infinite, x being variable? 
It is clear, intuitively, that the limit function cannot exhibit isolated 
points, as before, but must be represented by a connected curve. It 
would appear probable that this limit curve must consist of the con- 
tinuous branches of y = f (x) , together with the vertical segments which 
join / (x + 0) and / (x 0) at the points of 
discontinuity, that is, in our example, the 
curve would be shaped like a German m, as 
is shown in Fig. 83. The fact is, however, that 
the vertical part of the limit curve projects 
beyond / (x + 0) and / (x 0) , by a finite 
amount, so that the limit curve has the re- 
markable form sketched in Fig. 88. 

This little superimposed tower was noticed 
in the curves which the Michelson machine drew; in other words it 
was disclosed experimentally. At first it was ascribed to imperfec- 
tions in the ^ apparatus, but finally Gibbs recognized it as necessary. 
If D = |/ (% -f- o) / (x 0)| is, in general, the magnitude of the 
jump, then the length of the extension is, according to Gibbs: 

: 0.28D ^0.09#. 

As to the proof of this statement, it is sufficient to give it for a single 
discontinuous function, e.g., the one in our example, since all other 
functions with the same spring can be obtained from it by the addition 
of continuous functions. This proof is not very difficult. It results 

Fig. 88. 

Vol. 59 (189899), p. 200. Scientific papers II, p. 158. New York 1906. 

200 Analysis: The Goniometric Functions. 

immediately from consideration of the integral formula for S n (x) (see 
p. 196). Furthermore, if one draws a sufficient number of the approxi- 
mating curves one sees quite clearly how the Gibbs point arises. 

It would lead me too far afield if I were to consider further the 
many interesting niceties in the behaviour of the approximating curves. 
I am glad to refer you to the full and very readable article by Fejer 
in Vol. 64 (1907) of the Mathematische Annalen. 

With this I shall conclude the special discussion of trigonometric 
series in order to wander in a field which as to its content and its history 
is closely related to them. 

Excursus Concerning the General Notion of Function 

We must be all the more willing, in these lectures, to discuss the 
notion of function, since our school reform movement advocates giving 
this important concept a prominent place in instruction. 

If we follow again the historical development, we notice first that 
the older authors, like Leibniz and the Bernoullis, use the function 
concept only in isolated examples, such as powers, trigonometric func- 
tions, and the like. A general formulation is met first in the eighteenth 

1. With Euler, about 1750 (to use only .round numbers), we find 
two different explanations of the word function. 

a) In his Introductio he defines, as a function y of x , every analytic 
expression in x, i.e., every expression which is made up of powers, 
logarithms, trigonometric functions, and the like ; but he does not indicate 
precisely what combinations are to be admitted. Moreover, he had, 
already, the familiar division into algebraic and transcendental functions. 

b) At the same time, a function y (x) 
(see Fig. 89) was defined for him when- 
ever a curve was arbitrarily drawn 
(libero manus ductu) in an x , y coordi- 
nate system. 

^^ 2. Lagrange, about 1800, in his 
Fig. 89. Theorie des fonctions analytiquqs restricts 

the notion function, in comparison witlj 

Euler' s second definition, by confining it to so called analytic functions, 
which are defined by a power series in x. Modern usage has retained 
the words analytic functions with this same meaning, where, of course, 
one must recognize that this includes only a special class of the func- 
tions that really occur in analysis. Now a power series 

y = P(x) = a + a l x + a 2 x*+... 

defines a function primarily only within the region of its convergence, 
i.e., in a certain region around x = 0. A method was soon found, 


Excursus Concerning the General Notion of Function. 201 

however, for extending beyond this the region of definition for the 
function. If, say, x l (see Fig. 90) is within the region of convergence 
of P (x) , and if P (x) is resolved into a new series 

which proceeds according to powers of (x x-^ f it is possible that this 

may converge in a region extending beyond the first one, and so 

may define y in a larger field. A repetition of 

this process may extend the field still farther. 

This method of analytic continuation is well 

known to any one who is familiar with com- 

plex function theory. 

Notice, in particular, that every coefficient Fig. 90. 

in the power series P (x) , and therefore the 

entire function y is determined when the behavior of the function y 
along an arbitrarily small segment of the x axis is known, say in the 
neighborhood of x = 0. For then the values of all the derivatives of 
y are known for x = 0, and we know that 

y (o) = fl , /(o) = i , /'(o) = 2*2 , . . . 

Thus an analytic function, in the Lagrange sense, is determined through- 
out its entire course by the shape of an arbitrarily small segment. This 
property is completely opposed to the behavior of a function in the 
sense of Euler's second definition. There, any part of a curve can be 
continued at will. 

3. The further development of the function concept is due to 
J. J. Fourier, one of the numerous important mathematicians who 
worked in Paris at the beginning of the nineteenth century. His chief 
work is the Theorie analytique de la chaleur 1 which appeared in 1822. 
Fourier made the first communication, however, concerning his theories, 
to the Paris Academy in 1807- This work is the source of that far 
reaching method, so much used in mathematical physics today, which 
can be characterized as the resolution of all problems to the integration 
of partial differential equations with initial conditions, to a so called 
boundary+value problem. 

Fourier treated, in particular, the problem of heat conduction which, 
for a simple case, may be stated as follows. The boundary of a circular 
plate is kept at a constant temperature, e.g., one part at the freezing, 
the other at the boiling point (see Fig. 91). What stationary temperature 
is ultimately brought about by the resulting flow of heat? Boundary 
values are introduced here which can be assigned independently of each 
other at different parts of the boundary. Thus Euler's second definition 

1 Reprinted in Fourier, CEuvres, vol. I. Paris 1888. Translated into German 
by Weinstein. Berlin 1884. 

202 Analysis: The Goniometric Functions. 

of function comes appropriately into the foreground, as opposed to that 
of Lagrange. 

This definition is retained essentially by Dirichlet in the works which 
we mentioned (p. 197), except that it is translated into the language 
of analysis or, to use a modern term, it is arithmetized. This is in fact, 
necessary. For no matter how fine a curve be drawn, it can never 
define exactly the correspondence between the values of x and y. The 
stroke of the pen will always have a certain width, from which it follows 
that the lengths x and y which correspond to one another can be measured 
exactly only to a limited number of decimal places. 

Dirichlet formulated the arithmetic content of Euler's definition in 
the following way. If in any way a definite value of y is determined, 
corresponding to each value of x in a given interval, 
egrees ^ en y ^ ca ^ ec j a f unc tion of x. Although he announced 
this very general notion of a function, nevertheless he 
always thought primarily of continuous functions, or 
of such as were not all too discontinuous, as was done 
then quite generally. People considered complicated 
.100 degrees clusterings of discontinuities as thinkable, but they 
Fig. 91. hardly believed that they deserved much attention. 
This standpoint finds expression when Dirichlet speaks 
of the development into series of "entirely arbitrary functions" (just as 
Fourier had said "fonctions entierement arbitraires) even when he 
formulated very precisely his Dirichlet conditions, which must be satis- 
fied by all the functions he considered. 

5. We must now take account of the fact that at about this time, 
say around 1830, the independent development of the theory of 
functions of a complex argument began; and that in the next three 
decades it became the common property of mathematicians. This 
development was connected, above all, with the names Cauchy, Rie- 
mann, and Weierstrass. The first two start, as you know, from the 
partial differential equations which bear their names, and which must 
be satisfied by the real and imaginary parts u, v of the complex function 

while Weierstrass defines the function by means of a power series and 
the aggregate of its analytic continuations, so that he, in a sense, follows 

Now it is remarkable that this passage into the complex domain 
brings about an agreement and connection between the two function 
concepts considered above. I shall give a brief sketch of this. 

Let us put z = x + iy, and consider the power series 

(1) /(*) = u + iv = c + c r z + c 2 z* + - - - , 

Excursus Concerning the General Notion of Function. 203 

as converging for small \z\ so that, in the terminology of Weierstrass, 
it defines an element of an analytic function. We consider its values 
on a sufficiently small circle of radius r , about z = , which lies entirely 
within the region of convergence (see Fig. 92), i. e., we put z = x + iy 
= r (cos (p + i sin <p) in the power series, and we get 

/(*) = C Q + ^^(cosy + isin<p) + C 2 r 2 (cos2<p + isin2<p) 
If we separate the coefficients into real and imaginary parts: 

we get as the real part of / (z) 



u = u(<p) = - 

The sign of the imaginary part in the c was taken negative in order 

that all the signs should be positive. Thus the power series for / (z) 

yields for the values, on our circle, of the real part u t thought of as a 

function of the angle (p, a trigonometric series 

of exactly the former sort, whose coefficients z " ane 

are # , r v ot v , r v p v . 

Of course, these values u will be analytic 
functions of <p, in the sense of Lagrange, as long 
as the circle (r) lies entirely within the region of 
convergence of the power series (1). But if we 
allow it to coincide with the circle of conver- Fig. 92. 

gence of the series (1) which bounds its region 

of convergence, then the series (1) and consequently also the series (2) 
will not necessarily converge any longer. Meantime it can happen that 
the series (2) continues to converge, in which case the boundary values 
u (<p) cannot be analytic functions in the sense of Dirichlet. 

If we proceed conversely and assign to circle (r} an arbitrary distribu- 
tion of values u (<p) which satisfy only the conditions of Dirichlet, then 
they can be developed into a trigonometric series of the form (2), so 
that the*quantities ot , <x lf . . ., ft lf /? 2 , . . . and hence the coefficients of 
*the power series (1) (to within an arbitrary additive constant (i/? )/2) 
will be determined. It can be shown that this power series actually 
converges within the circle (r) and that the real part of the analytic 
function which it determines has the values u (q>) as boundary values 
on the circle (r) , or, to be more exact, that it approaches the value u (9?) 
whenever a position <p is approached for which u (<p) is continuous. 

The proofs of these facts are all contained in the investigations 
concerning the behavior of power series on the circle of convergence. 
I cannot, of course, give them here. But these remarks may serve to 

204 Analysis: The Goniometric Functions. 

show how, in this way, the Fourier-Dirichlet function concept and that 
of Lagrange merge into each other in that the arbitrariness in the 
behaviour of the trigonometric series u (<p) on the boundary of the 
circle is concentrated, for the power series, into the immediate neighbor- 
hood of the center. 

6. Modern science has not stopped with the formulation of these 
concepts. Science never rests, even though the individual investigator 
may become weary. During the last three decades mathematicians, 
taking a standpoint quite different from that of Dirichjet, have siezed 
upon functions having the greatest possible discontinuity, which, in 
particular, do not satisfy the Dirichlet conditions. The most remarkable 
types of function have been found, which contain the most disagreeable 
singularities "balled into horrid lumps". It becomes a problem then to 
determine how far the theorems which hold for "reasonable* ' functions 
still have validity for such abnormities. 

; S 7. In connection with this, there has arisen, finally, a still more far 
reaching generalization of the notion of function. Up to this time, a 
function was thought of as always defined at every position in the 
continuum made up of all the real or complex values of x, or at least 
at every position in an entire interval or region. But recently the theory 
of point sets, invented by G. Cantor, has made its way more and more 
to the foreground, in which the continuum of all % is only an obvious 
example of a set of points. From this new standpoint functions are 
being considered which are defined only for the positions x of some 
arbitrary set, so that in general y is called a function of x when to every 
element of a set x of things (numbers or points) there corresponds an 
element of a set y. 

Let me point out a difference between this newest development and 
the older one. The notions considered under headings 1. to 5. have 
arisen and have been developed with reference primarily to applications 
in nature. We need only think of the title of Fourier's work. But the 
newer investigations mentioned in 6. and 7. are the result purely of 
the love of mathematical research, which has taken no account whatever 
of the needs of natural phenomena, and the results have indeed found 
as yet no direct application. The optimist will think, of course, 'that the 
time for such application is bound to come. 

We shall now put our customary question as to how much of all 
this should be taken up by the schools. What should the teacher and 
what should the pupils know? 

In this connection I should like to say that it is not only excusable 
but even desirable that the schools should always lag behind the most 
recent advances of our science by a considerable space of time, certainly 
several decades; that, so to speak, a certain hysteresis should take place. 
But the hysteresis which actually exists at the present time is in some 

Excursus Concerning the General Notion of Function. 205 

respects unfortunately much greater. It embraces more than a century, 
in so far as the schools, for the most part, ignore the entire development 
since the time of Euler. There remains, therefore, a sufficiently large 
field for the work of reform. And what we demand in the way of reform 
is really quite modest, if you compare it with the present state of the 
science. We desire merely that the general notion of function, according 
to the one or the other of Euler 's interpretations, should permeate as 
a ferment the entire mathematical instruction in the higher schools. 
It should not, of course, be introduced by means of abstract definitions, 
but should be transmitted to the student as a living possession, by 
means of elementary examples, such as one finds in large number in 
Euler. For the teacher of mathematics, however, something more than 
this seems desirable, at least a knowledge of the elements of complex 
function theory; and although I should not make the same demand 
regarding the most recent concepts in the theory of point sets, still it 
seems very desirable that among the many teachers there should always 
be a small number who devote themselves to these things with the 
thought of independent work. 

I should like to add to these last remarks a few words concerning 
the important role that has been played in this entire development by 
the theory of trigonometric series. You will find extensive references 
to the literature of the subject in Burkhardt's Entwickelungen nach oszil- 
lierenden Funktionen (especially in chapters 2, 3> 7), that "giant report 11 , 
as his friends call it, which since 1901 has been appearing serially in 
volume 10 of the Jahresbericht der deulschen Mathematikervereinigung 1 . It 
combines, in more than 9000 references, an amount of pertinent literature 
such as you will hardly find elsewhere. 

The first to come upon the representation of general functions by 
means of trigonometric series was Daniel Bernoulli, the son of John 
Bernouilli. He noticed, about 1750, in his study of the acoustic problem 
of vibrating strings, that the general vibration of a string could be 
represented by the superposition of those sine vibrations which cor- 
responded to the fundamental tone and the overtones. That involves 
precisely, the development into a trigonometric series of the function 
vhich represents the form of the string. 

Although advances were soon made in knowledge of these series, 
itill no one really believed that arbitrary functions graphically given, 
:ould be represented by them. At bottom, here, there was an undefined 
presentiment of considerations which have become quite clear to us 
low through the theory of point sets. Perhaps one assumed, without, 

1 Completed in two half volumes as Heft 2 of this volume. Leipzig 1908. 

A short summary appears in the Enzyklopadie der mathematischen Wissen- 

ichaften, vol. 2. Burkhardt's report goes to 1850. The development from 1850 

on is sketched by Hilb and Riesz in their article in the Enzyklopadie, vol. 2, C 10.] 

206 Analysis: The Goniometric Functions. 

of course, being able to give precise expression to the feeling, that the 
"set" of all arbitrary functions, even if discontinuities are excluded, 
was greater than the "set" of all possible systems of numbers a^a^, 
a 2 , . . . , b lf b 2 , . . . , which represents the totality of trigonometric series. 
It is only the precise concepts of the modern theory of point sets that 
have cleared this up, and have shown that that judgment was false. Let 
me, at this place, elaborate somewhat this important point. It is easy 
to see that the entire course of a continuous function arbitrarily defined 
in a given interval, say from to 2 n t is completely known if one knows 
its values at all the rational positions of that interval (see Fig. 93)- 
For, since the set of these rational points is dense, we can effect an 

arbitrarily close approximation for any ir- 
rational position, in terms of function values 
at rational ones, so that, by virtue of the 
continuity of the function, the value of f(x) 
is known as the limit of the function values 
>x at -the approximating points. Furthermore, 
Fig. 93. we know that the set of all rational numbers 

is denumerable (see appendix II, p. 252), i. e., 

that they can be arranged in a series in which a definite first element 
is followed by a definite second, this by a definite third, and so on. 
From this it follows, however, that the assignment of the arbitrary 
continuous function means nothing more than the assignment of an 
appropriate denumerable set of constants the function values at the 
ordered rational points. But in the same way, by means, namely, of 
the denumerable series of constants # , a lt b lt a 2 , b 2 , . . ., we can assign 
a definite trigonometric series, so that the doubt as to whether the 
totality of continuous functions was, in the nature of things, essentially 
greater than that of the series, is groundless. Similar considerations hold 
for functions which are discontinuous but which satisfy the Conditions 
of Dirichlet. We shall have occasion later to give detailed consideration 
to these matters. 

The man who abruptly brushed aside all these misgivings was 
Fourier and it was just this which made him so significant in tfye history 
of trigonometric series. Of course, he did not base his conclusions on 
the theory of point sets, but he was the first one who had the courage 
to believe in the general power of series for purposes of representation. 
Fortified by this belief he set up a number of series by actual calculation, 
using characteristic examples of discontinuous functions, as we did a 
short time back. The proofs of convergence, as we have noted, were 
first given later, by Dirichlet, who, moreover, was a pupil of Fourier. 
This stand of Fourier's had a revolutionary effect. That it should be 
possible to represent by series of analytic functions such arbitrary 
functions as these, which obeyed in different intervals such entirely 

General Considerations in Infinitesimal Calculus. 207 

different laws, this was something quite new and unexpected to the 
mathematicians of that time. In recognition of the disclosure of this 
possibility, the name of Fourier was given to the trigonometric series 
which he employed, a name which has persisted to this day. To be 
sure every such personal designation implies a marked one-sidedness, 
even when it is not outright injustice. 

In conclusion, I must mention briefly a second accomplishment of 
Fourier. He considered, namely, the limiting case of the trigonometric 
series when the period of the function to be represented is allowed to 
become infinite. Since a function with an infinite period is simply a 
non periodic function, arbitrary along the entire % axis, this limiting 
case supplies a means of representing non periodic functions. The transi- 
tion is brought about by introducing a linear transformation of the 
argument of the series, which effects a representation of functions with 
a period / instead of 2 n, and then letting I become infinite. The series 
then goes over into the so called Fourier integral 


f(x) = [cp (v) cosvx + w (v) sinvx] dv , 

when <p (v) , y (v) are expressed in definite manner as integrals of the 
function / (x) from oo to + oo. The new thing here is that the index v 
takes continuously all values from to oo, not merely the values 0, 1, 
2, . . .; and that, correspondingly, <p (v)dv and ip (v)dv take the place 
of a v> b v . 

We shall now leave the elementary transcendental functions, which 
have hitherto been our chief concern in our remarks on analysis, and 
go over to a new concluding chapter. 

III. Concerning Infinitesimal Calculus Proper 

' Of course I shall assume that you all know how to differentiate and 
integrate, and that you have frequently used both processes. We shall 
be concerned here solely with more general questions, such as the logical 
and psychological foundations, instruction, and the like. 

i. General Considerations in Infinitesimal Calculus 
I should like to make a general preliminary remark concerning the 
range of mathematics. You can hear often from non mathematicians, 
especially from philosophers, that mathematics consists exclusively in 
drawing conclusions from clearly stated premises; and that, in this 
process, it makes no difference what these premises signify, whether they 
are true or false, provided only that they do not contradict one another. 
But a person who has done productive mathematical work will talk 
quite differently. In fact those persons are thinking only of the crystal- 
lized form into which finished mathematical theories are finally cast. 

208 Analysis: Concerning Infinitesimal Calculus Proper. 

The investigator himself, however, in mathematics, as in every other 
science, does not work in this rigorous deductive fashion. On the con- 
trary, he makes essential use of his phantasy and proceeds inductively, 
aided by heuristic expedients. One can give numerous examples of 
mathematicians who have discovered theorems of the greatest importance, 
which they were unable to prove. Should one, then, refuse to recognize 
this as a great accomplishment and, in deference to the above definition, 
insist that this is not mathematics, and that only the successors who 
supply polished proofs are doing real mathematics? After all, it is an 
arbitrary thing how the word is to be used, but no judgment of value 
can deny that the inductive work of the person who first announces 
the theorem is at least as valuable as the deductive work of the one who 
first proves it. For both are equally necessary, and the discovery is 
the presupposition of the later conclusion. 

It is precisely in the discovery and in the development of the 
infinitesimal calculus that this inductive process, built up without 
compelling logical steps, played such a great role; and the effective 
heuristic aid was very often sense perception. And I mean here im- 
mediate sense perception, with all its inexactness, for which a curve 
is a stroke of definite width, rather than an abstract perception which 
postulates a completed passage to the limit, yielding a one dimen- 
sional line. I should like to corroborate this statement by outlining 
to you how the ideas of the infinitesimal calculus were developed 

If we take up first the notion of an integral, we notice that it begins 
historically with the problem of measuring areas and volumes (quadra- 
ture and cubature). The abstract logical definition determines the 

f(x) dx, i.e., the area bounded by the curve y = /(#), the % 

axis, and the ordinates x = a , % = b , as the limit of the sum of narrow 
rectangles inscribed in this area when their number increases and their 
width decreases without bound. Sense perception, however, makes it 
natural to define this area, not as this exact limit, but simply as the 
sum of a large number of quite narrow rectangles. In fact, the necessary 
inexactness of the drawing would inevitably set bounds to the further 
narrowing of the rectangles (see Fig. 94). 

This naive method characterizes, in fact, the thinking of the greatest 
investigators in the early period of infinitesimal calculus. Let me men- 
tion, first of all, Kepler who in his Nova stereometria doliorum vinario- 
rum 1 was concerned with the volumes of bodies. His chief interest 
here was in the measuring of casks, and in determining their most suit- 
able shape. He took precisely the naive standpoint indicated above. 

1 Linz on the Danube, 1615. German in Ostwalds Klassikern, No. 165- Leipzig, 

General Considerations in Infinitesimal Calculus. 


He thought of the volume of the cask, as of every other body (see Fig. 95), 
as made up of numerous thin sheets suitably ranged in layers, and 
considered it as the sum of the volumes of these leaves, each of which 
was a cylinder. In a similar way he calculated the simple geometric 
bodies, e. g., the sphere. He thought of this as made up of a great 
many small pyramids with common vertex at the center (see Fig. 96). 
Then its volume, according to the well known formula for the pyramid, 
would be 7/3 times the sum of the bases of all the small pyramids. By 
writing for the sum of these little facets simply the surface of the sphere, 
or 4 n r*, he obtained 4 n r*/') , the correct formula for the volume. 

Fig. 94. 

Fig. 95- 

Fig. 96. 

Moreover, Kepler emphasizes explicitly the practical heuristic value of 
such considerations, and refers, so far as rigorous mathematical proofs 
are concerned, to the so called method of exhaustion. This method, which 
had been used by Archimedes, determines, for example, the area of the 
circle by following carefully the approximations to the area by means 
of inscribed and circumscribed polygons with an increasing number of 
sides. The essential difference between it and the modern method lies 
in the fact that it tacitly assumes, as self evident, the existence of a 
number which measures the area of the circle, whereas the modern 
infinitesimal calculus declines to accept this intuitive evidence, but has 
recourse to the abstract notion of limit and defines this number as the 
limit of the numbers that measure the areas of the inscribed polygons. 
Granted, however, the existence of this number, the method of ex- 
haustion is an exact process for approximating to areas by means of 
the known areas of rectilinear figures, one which satisfies rigorous 
modern demands. The method is, however, very tedious in many cases, 
and ill suited to the discovery of areas and volumes. One of Archimedes 
writings 1 , discovered by H. Heiberg in 1906, shows, in fact, that he did 
not use the method of exhaustion at all in his investigations. After 
he had first obtained his results by some other method, he developed 
the proof by exhaustion in order to meet the demands of that time as 
to rigor. For the discovery of his theorems he used a method which 
included considerations of the center of gravity and the law of the lever, 
and also of intuition, such as, for example, that triangles and parabolic 

1 Already referred to on p. 80. 
Klein, Elementary Mathematics. 



Analysis: Concerning Infinitesimal Calculus Proper. 

segments consist of series of parallel chords, or that cylinders, spheres, 
and cones are made up of series of parallel circular discs. 

Returning now to the seventeenth century, we find considerations 
analogous to those of Kepler in the book of the Jesuit Bonaventura 
Cavalieri: Geometria indivisibilibus continuorum nova quadam ratione 
promota 1 where he sets up the principle called today by his name: Two 
bodies have equal volumes if plane sections equidistant from their bases 
have equal areas. This principle of Cavalieri is, as you know, much used 
in our schools. It is believed there that integral calculus can be avoided 
in this way, whereas this principle belongs, in fact, entirely to the 
calculus. Its establishment by Cavalieri amounts precisely to this, that 
he thinks of both solids as built up of layers of thin leaves which, ac- 
cording to the hypothesis, are congruent in pairs, i.e., one of the bodies 
could be transformed into the other by translating its individual leaves 
(see Fig. 97) ; but this could not alter the volume, since this consists of 
the same summands before and after the translation. 

Naive sense perception leads in the same 
way to the differential quotient of a function, 
i. e., to the tangent to the curve. In this case, 
we can replace (and this is the way it was 
actually done) the curve by a polygonal line 
(see Fig. 98) which has on the curve a suffi- 
cient number of points, as vertices, taken close 
together. From the nature of our sense percep- 
tion we can hardly distinguish the curve from this aggregate of points 
and still less from the polygonal line. The tangent is now defined 
outright as the line joining two successive points, that is, as the 
prolongation of one of the sides of the polygon. 
From the abstract logical standpoint, this line 
remains only a secant, no matter how close 
together the points are taken; and the tangent 
is only the limiting position approached by the 
secant when the distance between the points 
approaches zero. Again, from this naive stand- 
point, the circle of curvature is thought of as the circle which passes 
through three successive polygon vertices, whereas exact procedure 
defines the circle of curvature as the limiting position of this circle 
when the three points approach each other. 

The force of conviction inherent in such naive guiding reflections is, 
of course, different for different individuals. Many and I include 
myself here find them very satisfying. Others, again, who are gifted 
only on the purely logical side, find them thoroughly meaningless and 
are unable to see how anyone can consider them as a basis for mathe- 

1 Bononiae, 1635. First edition, 1653. 

Fig. 97. 

Fig, 98. 

General Considerations in Infinitesimal Calculus. 

rnatical thought. Yet considerations of this sort have often formed 
the beginnings of new and fruitful speculations. 

Moreover, these naive methods always rise to unconscious importance 
whenever in mathematical physic, mechanics, or differential geometry 
a preliminary theorem is to be set up. You all know that they are 
very serviceable then. To be sure, the pure mathematician is not 
sparing of his scorn on these occasions. When I was a student it was 
said that the differential, for a physicist, was a piece of brass which he 
treated as he did the rest of his apparatus. 

In this connection, I should like to commend the Leibniz notation, 
the leading one today, because it combines with a suitable suggestion 
of nai've intuition, a certain reference to the abstract limit process which 
is implicit in the concept. Thus, the Leibniz symbol dy/dx, for the 
differential quotient, reminds one, first that it comes from a quotient; 
but the d, as opposed to the A which is the usual symbol for finite 
difference, indicates that something new has been added, namely, the 
passage to the limit. In the same way, the integral symbol / y dx sug- 
gests the origin of the integral from a sum of small quantities. However, 
one does not use the usual sign 2 for a sum, but rather a conventionalized 
5*, which indicates here that something new has entered the process 
of summation. 

We shall now discuss with some detail the logical foundation of 
differential and integral claculus, and at the same time consider it in 
its historical development. 

1. The principal idea, as the subject is taught, in general, at the 
university (I need only briefly to refresh your memory here) is that 
infinitesimal calculus is only an application of the general notion of limit. 
The differential quotient is defined as the limit of the quotient of 
corresponding finite increments of variable and function 

dy ,. Ay 
~ = lim ~- 

dx AX^ AX 

provided that this limit exists; and not at all as a quotient in which dy 
,and dx have an independent meaning. In the same way, the integral 
is defined as the limit of a sum: 

ydx = lim 
-i Axt=Q 

where the Axi are finite parts of the interval a^x^b, the % cor- 
responding arbitrary values of the function in that interval, and all 
iheAxi are to converge toward zero; but y dx does not have any actual 
significance as, say, a summand of a sum. These designations are 
retained for the reasons of expediency which we mentioned above. 

* It is remarkable that many are unaware that f has this meaning. 


212 Analysis: Concerning Infinitesimal Calculus Proper. 

2. The conception as we have thus characterized it is set forth in 
precise form by Newton himself. I refer you to a place in his principal 
work, the Philosophiae Naturalis Principia Mathematical of 1687-' "Ulti- 
mae rationes illae, quibuscum quantitates evanescunt, revera non sunt 
rationes quantitatum ultimarum, sed limites, ad quos quantitatum sine 
limite descrescentium rationes semper appropinquant, et quos propius 
assequi possunt, quam pro data quavis differentia, nunquam vero trans- 
gredi neque prius attingere quam quantitates diminuuntur in infinitum." 
Moreover, Newton avoids the infinitesimal calculus, as such, in the 
discussions in this work, although he certainly had used it in deriving 
his results. For, the fundamental work in which he developed his method 
of infinitesimal calculus was written in 1671, although it did not appear 
until 1736. It bears the title Methodus Fluxionum et Serierum Infini- 

In this, Newton develops the new calculus in numerous examples, 
without going into fundamental explanations. He makes connection 
here with a phenomenon of daily life which suggests a passage to a 
limit. If one considers, namely, a motion x = f (t) on the x axis in the 
time t, then every one has a notion as to what is meant by the velocity 
of this motion. If we analyze this motion it turns out that we mean 
the limiting value of the difference quotient Ax/ At. Newton made this 
velocity of x with respect to the time the basis of his developments. He 
called it the "fluxion" of x and wrote it #. He considered all the variables 
x, y as dependent on this fundamental variable t, the time. Accordingly 
the differential quotient dy/dx appears as the quotient of two fluxions 
y/x which we now should write more fully (dy/dt: dxjdt). 

3. These ideas of Newton were accepted and developed by a long 
series of mathematicians of the eighteenth century, who built up the 
infinitesimal calculus, with more or less precision, upon the notion of 
limit. I shall select only a few names: C. Maclaurin, in his Treatise of 
Fluxions*, which as a textbook certainly had a wide influence; then 
d'Alembert, in the great French Encyclopedie Methodique; and finally 
Kastner 4 , in Gottingen, in his lectures and books. Euler belongs pri- 
marily in this group although, with him, other tendencies also came 
to the front. 

4. It was necessary to fill out an essential gap in all these develop- 
ments, before one could speak of a consistent system of infinitesimal 
calculus. To be sure, the differential quotient was defined as a limit, 
but there was lacking a method for estimating, from it, the increment 

1 New edition by W. Thomson and H. Blackburn, Glasgow, 1871, p. 38i 

2 Newtoni, J., Opuscula Mathematica, philosophica, et philologica, vol.1, p. 29. 
Lausanne, 1744. 

a Edinburgh, 1742. 

4 Kastner, A. G., Anfangsgrunde der Analysis des Unendlichen, Gottingen, 1760. 

General Considerations in Infinitesimal Calculus. 


of the function in a finite interval. This was supplied by the mean value 
theorem] and it was Cauchy's great service to have recognized its funda- 
mental importance and to have made it the starting point accordingly 
of differential calculus. And it is not saying too much if, because of 
this, we adjudge Cauchy as the founder of exact infinitesimal calculus 
in the modern sense. The fundamental work in this connection, based 
on his Paris lectures, is his Resume des Lemons sur le Calcul Infinitesimal 1 , 
together with its second edition, of which only the first part, Lefons sur 
le Calcul Differentiel*, was published. 

The mean-value theorem, as you know, may be stated as follows. // 
a continuous function f (x) possesses a differential quotient f(x) every- 
where in a given interval, then there must be a point x + /M between x 
and % + h such that 

f(x + h)= f(x) + h*f(x + i)h) , (0 < * <1). 

Note here the appearance of that ft, peculiar to the mean value theorems, 
and which to beginners often seems so strange at first. Geometrically. 



Fig. 99. 

Fig. 100. 

the theorem is fairly obvious. It says, merely, that between the points 
x and x + h on the curve there is a point x + fth on the curve at 
which the tangent is parallel to the secant joining the points x and 
x + h (see Fig. 99). 

5. How can one give an exact arithmetic proof of the mean value 
theorem, without appealing to geometric intuition? Such a proof could 
only mean, of course, throwing the theorem back upon arithmetic de- 
finitions of variable, function, continuity etc., which would have to be 
set up in* advance in abstract, precise form. For this reason such a 
rigorous proof had to wait for Weierstrass and his followers, to whom, 
also, we owe the spread of the modern arithmetic concept of the number 
continuum. I shall try to give you the characteristic points of the 

In the first place, it is easy to make this theorem depend on the 
case where the secant is horizontal, i.e. / (x) = f (x + h) (see Fig. 100). 
One must then prove the existence of a place where the tangent is 

1 Paris, 1823- OEuvres completes, 2nd series, vol. 4, Paris, 1899- 

2 Paris, 1829. CEuvres completes, 2nd series, vol. 4, Paris, 1899- 

214 Analysis: Concerning Infinitesimal Calculus Proper. 

horizontal. To do this we can use the theorem of Weierstrass that 
every function which is constinubus throughout a closed interval actually 
reaches a maximum, and also a minimum value, at least once in that 
interval. Because of our assumption, one of these extreme values of 
our function must lie within the interval (x, x + h), provided we ex- 
clude the trivial case in which / (x) is a constant. Let us suppose that 
there is a maximum (the case of a minimum is treated in the same 
way) and that it occurs at the place x + &h. It follows that / (x) 
cannot have larger values, either to the right or to the left, i.e., the 
difference quotient to the right is negative, or zero, and to the left, 
positive or zero. Since the differential quotient exists, by hypothesis, 
at every place in the interval, its value at x + till can be looked upon 
as the limit of values which are either not positive or not negative, 
according as one thinks of it as a progressive or a regressive derivative. 
Therefore it must have the value zero, the tangent at x = $h is hori- 
zontal, and the theorem is proved. 

The scientific mathematics of today is built upon the series of 
developments which we have been outlining. But an essentially different 
conception of infinitesimal calculus has been running parallel with this 
through the centuries. 

1. What precedes harks back to old metaphysical speculations con- 
cerning the structure of the continuum according to which this was 
made up of ultimate indivisible infinitely small parts. There were already, 
in ancient times, suggestions of these indivisibles and they were widely 
cultivated by the scholastics and still further by the Jesuit philosophers. 
As a characteristic example I recall the title of Cavalieri's book, men- 
tioned on p. 210 Geometria Indivisibilibus Continuomm Promota, which 
indicates its true nature. As a matter of fact, he considers intuitive 
mathematical approximation in a secondary way only. He actually 
considers space as consisting of ultimate indivisible parts, the "indivisi- 
bilia". In this connection it would be interesting and important to 
know the various analyses to which the notion of the continuum has 
been subjected in the course of centuries (arid milleniums). 

2. Leibniz, who shares with Newton the distinction of having in- 
vented the infinitesimal calculus, also made use of such ideas. The 
primary thing for him was not the differential quotient thought of as 
a limit. The differential dx of the variable x had for him actual existence 
as an ultimate indivisible part of the axis of abscissas, as a quantity 
smaller than any finite quantity and still not zero ("actually*' infinitely 
small). In the same way the differentials of higher order d*x, d*x, . . . 
are defined as infinitely small quantities of second, third, . . . order, 
each of which is "infinitely small in comparison with the preceding". 
Thus one had a series of systems of qualitatively different magnitudes, 
According to the theory of indivisibles, the area bounded by the curve 

General Considerations in Infinitesimal Calculus, 215 

y = y (x) and the axis of abscissas is the direct sum of all the individual 
ordinates. It is because of this view that Leibniz, in his first manuscript 
on integral calculus (1675), writes jy and not fydx. 

This point of view, however, is by no means the only one which 
interested Leibniz. Sometimes he uses the notion of mathematical 
approximation, where, for example, the differential dx is a finite segment 
but so small that, for that interval, the curve is not appreciably different 
from the tangent. The above metaphysical speculations are surely only 
idealizations of these simple psychological facts. 

But there is a third direction for the mathematical ideas of Leibniz, 
one which is especially characteristic of him. It is his formal point of 
view. I have frequently reminded you that we can look upon Leibniz 
as the founder of formal mathematics. His thought here is as follows. 
It makes no difference what meaning we attach to the differentials, 
or whether we attach any meaning whatever to them. If we define 
appropriate rules of operation for them, and if we employ these rules 
properly, it is certain that something reasonable and correct will result. 
Leibniz refers repeatedly to the analogy with complex numbers, con- 
cerning which he had corresponding notions. As to these rules of ope- 
ration for differentials he was concerned chiefly with the formula 

The mean value theorem shows that this is correct only if one writes 
/' (x + & dx) instead of /' (x) ; but the error which one commits by 
writing /'(#) outright is infinitely small, of higher (second) order, and 
such quantities are to be neglected (this is the most important formal 
rule) in operations with differentials. 

The chief publications of Leibniz are contained in that famous first 
scientific journal, the Ada Eruditorum 1 ', in the years 1684, 1685, and 
1712. In the first volume, you find, under the title Nova methodus pro 
maximis et minimis (p. 467 et seq.), the very first publication concerning 
differential calculus. In this Leibniz merely develops the rules for 
differentiation. The later works give also expositions of principles, where 
preference, is given to the formal standpoint. In this connection, the 
sjiort article of the year 1712 2 , one of the last years of his life, was 
especially characteristic. In this he speaks outright of theorems and 
definitions which are only "tolemnter vera" or French "passables" : 
"Rigorem quidem non sustinent, habent tamen usum magnum in calcu- 
lando et ad artem inveniendi universalesque conceptus valent." He 
has reference here to complex numbers as well as to the infinite. If 

1 Translated, in part, in Ostwalds Klassikern No. 1 62. Edited by G. Kowalewski, 
Leipzig, 1908. Also in Leibniz, Mathematische Schriften. Edited by K. J. Ger- 
hardt, from 1849 on. 
, a Observatio . . .; et de vero sensu methodi infinitesimalis, p. 167 169- 

21 6 Analysis: Concerning Infinitesimal Calculus Proper. 

we speak, perhaps, of the infinitely small, then "commoditati expressio- 
nis seu breviloquio mentalis inservimus, sed non nisi toleranter vera 
loquimur, quae explicatione rigidantur." 

3. From Leibniz as center the new calculus spread rapidly over the 
continent and we find each of his three points of view represented. 
I must mention here the first textbook of differential calculus that ever 
appeared, the Analyse des Infiniment Petits pour V Intelligence des 
Courbes 1 by Marquis de T Hospital, a pupil of Johann Bernoulli, who 
for his part, had absorbed the new ideas from Leibniz with surprising 
speed and had himself published the first textbook on the integral 
calculus 2 . Both books represent the point of view of mathematics of 
approximation. For example, a curve is thought of as a polygon with 
short sides, a tangent as the prolongation of one of these sides. In 
Germany, the differential calculus according to Leibniz was spread 
widely by Christian Wolff, of Halle, who published the contents of his 
lectures in Elementa matheseos universal*. He introduces the differentials 
of Leibniz immediately, at the beginning of the differential calculus, 
although he emphasizes particularly that they have no actual equivalent 
of any kind. And, indeed, as an aid to our intuition he develops his 
views concerning the infinitely small in a manner which savors thoroughly 
of mathematics of approximation. Thus he says, by way of example, 
that for purposes of practical measurement, the height of a mountain 
is not noticeably changed by adding or removing a particle of dust. 

4. You will also frequently find the metaphysical view which ascribes 
an actual existence to the differentials. It has always had supporters, 
especially on the philosophic side, but also among mathematical physi- 
cists. One of the most prominent here is Poisson, who, in the preface 
to his celebrated Traite de Mecanique*, expressed himself strongly to 
the effect that the infinitely small magnitudes are not merely an aid 
in investigation but that they have a thoroughly real existence. 

5. Due probably to the philosophic tradition, this concept went 
over into textbook literature and plays a marked role there even today. 
As an example, I mention the textbook by Liibsen Einleitung in die 
Infinitesimalrechnung* which appeared first in 1855 and which had for 
a long time an extraordinary influence among a large part of the public. 
Everyone, in my day, certainly had Lubsen's book in his hand, either 
when he was a pupil, or later, and many received from it the first 

1 Paris, 1696; second edition, 1715- 

[ 2 Translated in Ostwalds Klassikern No. 194. Edited by G. Kowalewski. 
Job. Bernoulli's Differentialrechnung was discovered and discussed a short time 
ago by P. Schafheitlin. Verhandlungen der Naturforscher-Gesellschaft in Basel, 
vol. 32 (1921).] 

3 Appeared first in 1710. Editio nova Hallae, Magdeburgiae, 1742, p. 545. 

4 Part I, second edition, p. 14. Paris, 1833- 

5 Eighth edition, Leipzig, 1899- 

General Considerations in Infinitesimal Calculus. 217 

stimulation to further mathematical study. Liibsen defined the diffe- 
rential quotient first by means of the limit notion; but along side of 
this he placed (after the second edition) what he considered to be the 
true infinitesimal calculus a mystical scheme of operating with infinitely 
small quantities. These chapters are marked with an asterisk to indicate 
that they bring nothing new in the way of result. The differentials are 
introduced as ultimate parts which arise, for example, by continued 
.halving of a finite quantity an infinite, non assignable number of times ; 
and each of these parts "although different from absolute zero is never- 
theless not assignable, but an infinitesimal magnitude, a breath, an 
instant". And then follows an English quotation: "An infinitesimal is 
the spirit of a departed quantity" (p. 59, 60). Then in another place 
(p. 76): "The infinitesimal method is, as you see, very subtle, but 
correct. If this is not manifest from what has preceded, together with 
what follows, it is the fault only of inadequate exposition." It is cer- 
tainly very interesting to read these passages. 

As companion piece to this I put before you the sixth edition of 
the widely used Lehrbuch der Experimentalphysik by Wiillner 1 . The 
first volume contains a brief preliminary exposition of infinitesimal 
calculus for the benefit of those students of natural science or medicine 
who have not acquired, at the gymnasium, that knowledge of calculus 
which is indispensable for physics. Wiillner begins (p. 31) with the 
explanation of the meaning of the infinitely small quantity dx t then 
follows with the explanation for the second differential d z x, which, of 
course, is more difficult. I urge you to read this introduction with the 
eye of the mathematician and to reflect upon the absurdity of sup- 
pressing infinitesimal calculus in the schools because it is too difficult, 
and then of expecting a student in his first semester to gain an under- 
standing of it from this ten page presentation, which is not only far from 
satisfying, but very hard to read! 

The reason why such reflections could so long hold their place 
abreast of the mathematically rigorous method of limits, must be sought 
probably in the widely felt need of penetrating beyond the abstract 
ogical formulation of the method of limits to the intrinsic nature of 
xmtinuous magnitudes, and of forming more definite images of them 
:han were supplied by emphasis solely upon the psychological moment 
tfhich determined the concept of limit. There is one formulation which 
is characteristic, which is due, I believe, to the philosopher Hegel, and 
which formerly was frequently used in textbooks and lectures. It 
declares that the function y = / (x) represents the being, the differential 
quotient dy[dx, however, the becoming, of things. There is assuredly 
something impressive in this, but one must recognize clearly that such 

1 Leipzig, 1907. 

218 Analysis: Concerning Infinitesimal Calculus Proper. 

words do not promote further mathematical development because this 
must be based upon precise concepts. 

In the most recent mathematics, "actually" infinitely small quantities 
have come to the front again, but in entirely different connection, 
namely in the geometric investigations of Veronese and also in Hilbert's 
Grundlagen der Geometric 1 * The guiding thought of these investigations 
can be stated briefly as follows: A geometry is considered in which 
x = a (a an ordinary real number) determines not only one point on. 
the x axis, but infinitely many points, whose abscissas differ by finite 
multiples of infinitely small quantities of different orders TJ, f, . . . 
A point is thus determined only when one assigns 

x = a + by + c + , 

where a, 6, c, . . . are ordinary real numbers, and the 17, , . . . actually 
infinitely small quantities of decreasing orders. Hilbert uses this guiding 
idea by subjecting these new quantities i] , ?, . . . to such axiomatic 
assumptions as will make it evident that one can operate with them 
consistently. To this end it is of chief importance to determine appro- 
priately the relation as to. size between x and a second quantity x l = a t 
+ bitf + Cif + . The first assumption is that x > or < x l if 
a > or < ! ; but if a = a l , the determination as to size rests with the 
second coefficient, so that x^x l according as b ^ b^\ and if, in addition, 
b &! , the decision lies with the c , etc. These assumptions will be 
clearer to you if you refrain from attempting to associate with the 
letters any sort of concrete representation. 

Now it turns out that, after imposing upon these new quantities 
this rule, together with certain others, it is possible to operate with 
them as with finite numbers. One essential theorem, however, which 
holds in the system of ordinary real numbers, now loses its validity, 
namely the theorem : Given two positive numbers e , a, it is always possible 
to find a finite integer n such that n e> a, no matter how small e is 
nor how large a may be. In fact, it follows immediately from the above 
definition that an arbitrary finite multiple n 17 of r) is smaller than 
any positive finite number a, and it is precisely this property that 
characterizes the ij as an infinitely small quantity. In the same way 
n < YI , that is, is an infinitely small quantity of higher order than r\ . 

This number system is called non-Archimedean. The above theorem 
concerning finite numbers is called, namely, the axiom of Archimedes, 
because he emphasized it as an unprovable assumption, or as a funda- 
mental one which did not need proof, in connection with the numbers 
which he used. The denial of this axiom characterizes the possibility 
of actually infinitely small quantities. The name Archimedean axiom, 
however, like most personal designations, is historically inexact. Euclid 

1 Fifth edition, Leipzig, 1922. 

/General Considerations in Infinitesimal Calculus. 219 

gave prominence to this axiom more than half a century before Archi- 
medes ; and it is said not to have been invented by Euclid, either, but, 
like so many of his theorems, to have been taken over from Eudoxus 
of Knidos. The study of non-Archimedean quantities 1 , which have 
been used especially as coordinates in setting up a non-Archimedean 
geometry, aims at deeper knowledge of the nature of continuity and 
belongs to the large group of investigations concerning the logical de- 
pendence of different axioms of ordinary geometry and arithmetic. For 
this purpose, the method is always to set up artificial number systems 
for which only a part of the axioms hold, and to infer the logical in- 
dependence of the remaining axioms from these. 

The question naturally arises whether, starting from such number 
systems, it would be possible to modify the traditional foundations of 
infinitesimal calculus, so as to include actually infinitely small quantities 
in a way that would satisfy modern demands as to rigor; in other words, 
to construct a non-Archimedean analysis. The first and chief problem 
of this analysis would be to prove the mean-value theorem 

from the assumed axioms. I will not say that progress in this direction 
is impossible, but it is true that none of the investigators who have 
busied themselves with actually infinitely small quantities have achieved 
anything positive. . . 

I remark for your orientation that, sincy Cauchy's time, the words 
infinitely small are used in modern textbooks in a somewhat changed 
sense. One never says, namely, that a quantity is infinitely small, but 
rather that it becomes infinitely small; which is only a convenient ex- 
pression implying that the quantity decreases without bound toward zero. 

We must bear in mind the reaction which was evoked by the use 
of infinitely small quantities in infinitesimal calculus. People soon 
sensed the mystical, the unproven, in these ideas, and there arose often 
a prejudice, as though the differential calculus were a particular philo- 
sophical system which could not be proved, which could only be believed 
or, to put it bluntly, a fraud. One of the keenest critics, in this sense, 
^was the philosopher Bishop Berkeley, who in the little book The Analyst* 
assailed in an amusing manner the lack of clearness which prevailed 
in the mathematics of his time. Claiming the privilege of exercising the 
same freedom in criticizing the principles and methods of mathematics 
"which the mathematicians employed with respect to the mysteries of 
religion", he launched a violent attack upon all the methods of the new 

t 1 The so-called horn-shaped angles, known already to Euclid, are examples 
of non- Archimedean quantities. Compare also the excursus, in the second volume 
of this work, in connection with the critique of Euclid's Elements.] 
2 London, 1734. 

220 Analysis: Concerning Infinitesimal Calculus Proper. 

analysis, the calculus with fluxions as well as the operation with diffe- 
rentials. He came to the conclusion that the entire structure of analysis 
was obscure and thoroughly unintelligible. 

Similar views have often maintained themselves even up to the 
present time, especially on the philosophical side. This is due, perhaps, 
to the fact that acquaintance here is confined to the operation with 
differentials; the rigorous method of limits, a rather recent development, 
has not been comprehended. As an example, let me quote from Bau- 
mann's Raum, Zeit und Mathematik 1 which appeared in the sixties: 
"Thus we discard the logical and metaphysical justification, which 
Leibniz gave to calculus, but we decline to touch this calculus itself. 
We look upon it as an ingenious invention which has turned out well 
in practice; as an art rather than a science. It cannot be constructed 
logically. It does not follow from the elements of ordinary mathe- 
matics . . ." 

This reaction against differentials accounts also for the attempt by 
Lagrange, already mentioned, in his Theorie des Fonctions Analytiques, 
published in 1 797, to eliminate from the theory not only infinitely small 
quantities, but also every passage to the limit. He confined himself, 
namely, to those functions which are defined by power series 

and he defines formally the "derived function /' ' (x)" (he avoids charac- 
teristically the expression differential quotient and the sign dy/dx) by 
means of a new power series 

Consequently he talks of derivative calculus instead of differential calculus. 
This presentation, of course, could not be permanently satisfactory. 
In the first place, the concept of function used here is, as we have 
shown, much too limited. More than that, however, such thoroughly 
formal definitions make a deeper comprehension of the nature of the 
differential coefficient impossible, and take no account of what we called 
the psychological moment they leave entirely unexplained just f why one 
should be interested in a series obtained in such a peculiar way. Finally , t 
one can get along without giving any thought to a limit process only 
by disregarding entirely the convergence of these series and the question 
within what limits of error they can be replaced by finite sums. As soon 
as one begins a consideration of these problems, which is essential, of 
course, for any actual use of the series, it is necessary to have recourse 
precisely to that notion of limit, the avoidance of which was the purpose 
of inventing the system. 

1 Vol. 2, p 55- Berlin, 1869. 

General Considerations in Infinitesimal Calculus. 221 

It would be fitting, perhaps, to say a few words about the differences 
of opinion concerning the foundations of calculus, as these come up, 
even today, beyond the narrow circle of professional mathematicians. 
I believe that we can often find here the preliminary conditions for 
understanding, in considerations very similar to those which we set forth 
respecting the foundations of arithmetic (p. 13). In every branch of 
mathematical knowledge one must separate sharply the question as to 
the inner logical consistency of its structure from that as to the justi- 
fication for applying its axiomatically and (so to speak) arbitrarily 
formulated notions and theorems to objects of our external or internal 
perception. George Cantor 1 makes the distinction, with reference to 
whole numbers, between immanent reality, which belongs to them by 
virtue of their logical definability, and transient reality, which they 
possess by virtue of their applicability to concrete things. In the case 
of infinitesimal calculus, the first problem is completely solved by means 
of those theories which the science of mathematics has developed in 
logically complete manner (through the use of the concept of limit). 
The second question belongs entirely to the theory of knowledge, and 
the mathematician contributes only to its precise formulation when he 
separates from it and solves the first part. No pure mathematical work 
can, from its very nature, supply any immediate contribution to its 
solution. (See the analogous remarks on arithmetic, p. 13 et seq.) All 
disputes concerning the foundations of infinitesimal calculus labor under 
the disadvantage that these two entirely different phases of the problem 
have not been sharply enough separated. In fact, the first, the purely 
mathematical part, is established here precisely as in all other branches 
of mathematics, and the difficulties lie in the second, the philosophical 
part. The value of investigations which press forward in this second 
direction takes on especial importance in view of these considerations; 
but it becomes imperative to make them depend upon exact knowledge 
of the results of the purely mathematical work upon the first problem. 

I must conclude with this excursus our short historical sketch of 
the development of infinitesimal calculus. In it I was obliged of course 
to confine myself to an emphasis of the most important guiding notions. 
It shoulcl be extended, naturally, by a thorough-going study of the 
entire literature of that period. You will find many interesting references 
in the lecture given by Max Simon at the Frankfurt meeting of the 
natural scientists of 1896: Zur Geschichte und Philosophie der Differential- 

If we now examine, finally, the attitude towards infinitesimal 
calculus in school instruction, we shall see that the course of its historical 
development is mirrored there to a certain extent. In earlier years, 

1 Mathematische Annalen, vol. 21 (1883), p. 562. 

222: Analysis: Concerning Infinitesimal Calculus Proper. 

where infinitesimal calculus was taught in the schools, there was by no 
means a clear notion of its exact scientific structure as based on the 
method of limits. 'At least this was manifest in the textbooks, and it 
was doubtless the same in the schools. This method cropped up in a 
vague way at most, whereas operations with infinitely small quantities 
and sometimes also derivative calculus, in the sense of Lagrange, came 
to the front. Such instruction, of course, lacked not only rigor but 
intelligibility as well, and it is easy to see why a marked aversion arose 
to the treatment of infinitesimal calculus at all in the schools. This 
culminated in the seventies and eighties in an official order forbidding 
this instruction even in the "real" institutions. 

To be sure this did not entirely prevent (as I indicated earlier) the 
using of the method of limits in the schools, where it was necessary one 
merely avoided that name, or one even thought sometimes that some- 
thing else was being taught. I shall mention here only three examples 
which most of you will recall from your school days. 

a) The well known calculation of the perimeter and the area of the 
circle by an approximation which uses the inscribed and circumscribed 
regular polygons is obviously nothing but an integration. It was em- 
ployed, even in ancient times, and was used particularly by Archimedes ; 
in fact, it is owing to its classical antiquity that is has been retained 
in the schools. 

b) Instruction in physics, and particularly in mechanics, necessarily 
involves the notions of velocity and acceleration, and their use in various 
deductions, including the laws of falling bodies. But the derivation of 
these laws is essentially identical with the integration of the differential 
equation z" = g by means of the function z = \ gt 2 + at + b, where 
a, b are constants of integration. The schools must solve this problem, 
under pressure of the demands of physics, and the means which they 
employ are more or less exact methods of integration, of course disguised. 

c) In many North German schools the theory of maxima and minima 
was taught according to a method which bore the name of Schellbach, 
the prominent mathematical pedagogue of whom you all must have 
heard. According to this method one puts 

in order to obtain the extremes of the function y = / (x) , But that is 
precisely the method of differential calculus, only that the word differen- 
tial quotient is not used. It is certain that Schellbach used the above 
expression only because differential calculus was prohibited in the 
schools and he nevertheless did not want to miss these important 
notions. His pupils, however, took it over unchanged, called it by his 
name, and so it came about that methods which Fermat, Leibniz, and 

Taylor's Theorem. 

Newton had possessed were put before the pupils under the name of 

Let me now indicate, finally, the attitude toward these things of 
our reform tendency, which is now gaining ground more and more in 
Germany, as well as elsewhere, especially in France, and which we hope 
will control the mathematical instruction of the next decades. We 
desire that the concepts which are expressed by the symbols y = f (x) , 
dy/dx, fydx be made familiar to pupils , under these designations; not, 
indeed, as a new abstract discipline, but as an organic part of the total 
instruction; and that one advance slowly, beginning with the simplest 
examples. Thus one might begin, with pupils of the age of fourteen 
and fifteen, by treating fully the functions y = ax + b (a,b definite 
numbers) and y = x 2 , drawing them on cross-section paper, and letting 
the concepts slope and area develop slowly. But one should hold to 
concrete examples. During the next three years this knowledge could 
be gathered together and treated as a whole, the result being that the 
pupils would come into complete possession of the beginnings of in- 
finitesimal calculus. It is essential here to make it clear to the pupil 
that he is dealing, not with something mystical, but with simple things 
that anyone can understand. 

The urgent necessity of such reforms lies in the fact that they are 
concerned with those mathematical notions which govern completely 
the applications of mathematics which are being made today in every 
possible field, and without which all studies at the university, even the 
simplest studies in experimental physics, are suspended in mid air. 
I can be content with these few hints, chiefly because this subject is 
fully discussed in Klein-Schimmack (referred to on p. 3). 

In order to supplement these general considerations with something 
which again is concrete I shall now discuss in some detail an especially 
important subject in infinitesimal calculus. 

2. Taylor's Theorem 

I shall proceed here in a manner analogous to the plan I followed 
with trigonometric series. I shall depart, namely, from the usual 
.treatment in the textbooks by bringing to the foreground the finite 
series, so important in practice, and by aiding the intuitive grasp of 
the situation by means of graphs. In this way it will all seem elementary 
and easily comprehensible. 

We begin with the question whether we can make a suitable appro- 
ximation to an arbitrary curve y = / (x) , for a short distance, by means 
of curves of the simplest kind. The most obvious thing is to replace 
the curve in the neighborhood of a point x = a by its rectilinear tangent 

224 Analysis: Concerning Infinitesimal Calculus Proper. 

just as, in physics and in other applications, we often discard the higher 
powers of the independent variable in a series development (see Fig. 101). 
In a similar manner we can obtain better approximations by making 
use of parabolas of second, third, . . . order 

y = A + Bx + Cx*, y = A+Bx + Cx* + Dx*> . . . 

or, in analytic terms, by using polynomials of higher degree. Polynomials 

are especially suitable because they are 
so easy to calculate. We shall give all 
these curves a special position, so that 
at the point x = a they lie as close as 
possible to the curve, i.e., so that they 
shall be parabolas of osculation. Thus 
the quadratic parabola will coincide with 
y = f (x) not only in its ordinate but also 
in its first and second derivatives (i.e., 

it will "osculate"). A simple calculation shows that the analytic ex- 
pression for the parabola having osculation of order n will be 

y = /M + ^(*-.)+^ 

(n = l,2,3, .-.) 
and these are precisely the first n + 1 terms of Taylor's series. 

The investigation as to whether and how far these polynomials 
represent usable curves of approximation will be started by a some- 
what experimental method, such as we used in the case (p. 194) of the 
trigonometric series. I shall show you a few drawings of the first 
osculating parabolas of simple curves, which were made 1 by Schimmack. 
The first are the four following functions, all having a singularity at 
x = \ , drawn with their parabolas of osculation at x = (see Figs. 102, 
103, 104, 105). 

1. lOg(1+*) W X- ~+ y- + .-. , 

X v2 v3 

2. (1+*)* *M+ 8 -+ J6- + -, 

3. (1+tf)- 1 ^!- x+ op y* H 

4. (1 + #)- 2 ^l 2x + 3# 2 4* 3 H 

In the interval ( 1 , +1) the parabolas approach the original curve 
more and more as the order increases ; but to the right of x = +1 they 
deviate from it increasingly, now above, now below, in a striking way. 

1 Four of these drawings accompanied Schimmack's report on the Gottingen 
Vacation Course, Easter, 1908: Uber die Gestaltung des mathematischen Unter- 
richts im Sinne der neueren Reformideen, Zeitschrift fur den Mathematischen und 
naturwissenschaftlichen Unterricht, vol. 39 (1908), p. 513; also separate reprints. 
Leipzig, 1908. 

Taylor's Theorem. 


At the singular point x = \ , in Cases 1,3,4, where the original 
function becomes infinite, the ordinates of the successive parabolas are 
increasingly large. In Case 2, where the branch of the original curve 
which appears, in the drawing, ends in # = 1 at a vertical tangent, 

Fig. 102. 

F"ig. 104. 

all the parabolas extend beyond this point but approach the original 
curve more and more at x = \ , by becoming ever steeper. At the 
point x = +1 , symmetrical to % = 1 , the parabolas in the first two 
cases approach the original curve more and more closely. In Case 3, 
their ordinates are alternately equal to 1 and , while that of the original 
curve has the value . In Case 4, finally, the ordinates increase in- 
definitely with the order, and alternate in sign. 

Klein, Elementary Mathematics. 1 5 


Analysis: Concerning Infinitesimal Calculus Proper. 

We shall examine, now, sketches of the osculating parabolas of 

two integral transcendental functions (see Fig. 106, 107) 

v 2 /8 


s , 

6. si 

3-1-^ --.-.. 

You notice that as their order increases, the parabolas give usable 
aproximations to the original curve for a greater and greater interval. 
It is especially striking in the case of sin x how the parabolas make 
the effort to share more and more oscillations with the sine curve. 
I call your attention to the fact that the drawing of such curves 
in simple cases is perhaps suitable material even for the schools. After 

we have thus assembled our experimental 
I 1 material we must consider it mathematically. 


Fig. 106. 

Fig. 107- 

The first question here is the extremely important one in practice as 
to the closeness with which the w-th parabola of osculation represents 
the original curve. This implies an estimate of the remainder for the 
values of the ordinate, and is connected naturally with the passage 
of n to infinity. Can the curve be represented exactly, at least for a part 
of its course, by an infinite power series? 

It will be sufficient to state the commonest of the theorems con- 
cerning the remainder: 

*/ -r j, / \i -r . T (n- 1)| ' v 

The proof of the theorem is given in all the books and I shall revert 
to it later, anyway, from a more general standpoint. The theorem is: 
There is a value ( between a and x such that R n can be represented in the 

*(*) = 


' /<">(?), 

(a <<*).. 

Taylor's Theorem. 


Fig. 108. 

The question as to the justification of the transition to an infinite 
series is now reduced to that as to whether this R n (x) has the limit 
or not when n becomes infinite. 

Returning to our examples, it appears, as you can verify by reading 
anywhere, that in Cases 5 and 6 the infinite series converges for all 
values of x. In Cases 1 to 4, it turns out that the series converges, 
between \ and +1, to the original function, but that it diverges 
outside this interval. For x = \ we have, in Case 2, convergence to 
the function value; in Cases 1, 3, 4, the limiting value of the series as 
well as that of the function is infinite, so that one could speak of con- 
vergence here also, but it is not customary to use this word with a 
series that has a definitely infinite limit. For y 

x = +1, finally, we have convergence in the 
first two examples, divergence in the last two. 
All this is in fullest agreement with our graphs. 

We may now raise the question, as we 

did with the trigonometric series, as to the 

limiting positions toward which the approxi- 
mating parabolas converge, thought of as com- 
plete curves. They cannot, of course, break 
off suddenly at x = i 1 . For the case of 
log (1 + x) I have sketched for you the limit 
curve (Fig. 108). The even and odd parabolas 
have different limiting positions, (indicated in the figure by dashes 
and dots) which consist of the logarithm curve between 1 and +1 
together with the lower and upper portions, respectively, of the 
vertical line x = + 1 . The other three cases are similar. 

The theoretical consideration of Taylor's series cannot be made com- 
plete without going over to the complex variable. It is only then that 
one can understand the sudden ceasing of the power series to converge 
at places where the function is entirely regular. To be sure, one might 
be satisfied, in the case of our examples, by saying that the series 
cannot converge any farther to the right than to the left, and that the 
convergence must cease at the left because of the singularity at x = \. 
But such reasoning would not fit a case like the following. The Taylor's 
series development for the branch of tan" 1 * which is regular for all 
real x 

tan- 1 *^* " + ~E h"- 

converges only in the interval ( 1, + 1), and the parabolas of oscula- 
tion converge alternately to two different limiting positions (see Fig. 109)^ 
The first consists, in the figure, of the long dotted parts of the vertical 
lines x = +1, #= l together with the portion of the inverse tangent 
curve lying between these verticals. The second limiting position is 



Analysis: Concerning Infinitesimal Calculus Proper. 

Fig. 109- 

obtained from the first by taking the short dotted parts of the vertical 
lines instead of the long dotted parts. The convergence is toward the 

first of these limit curves when we take 
an odd number of terms in the series, 
toward the second when we take an 
even number. In the figure, the long dott- 
ed curve represents y = x # 3 /3 + # 5 /5 , 
the short dotted curve is y = x # 3 /3 . 
The sudden cessation of convergence at 
~~jc the thoroughly regular points x = 1 is 
incomprehensible if we limit ourselves 
to real values of x and notice the be- 
havior of the function. The explanation 
is to be found in the important theorem 
on the circle of convergence, the most 
beautiful of Cauchy's function-theoretic 
achievements, which can be stated as 
follows. // one marks on the complex 
x plane all the singular points of the analytic junction f (x) , when f (x) is 
single-valued, and on the Riemann surface belonging to f (x) when f (x) is 
many -valued, then the Taylor's series corresponding 
to a regular point x = a converges inside the largest 
circle about a which has no singular point in its 
interior (i.e., so that at least one singular point 
lies on its circumference). The series converges 
for no point outside this circle (see Fig. 110). 
Now our example tan" 1 A; has, as you know, 
singularities at x = i, and the circle of con- 
vergence of the development in powers of x is 
consequently the unit circle about x = . The 
convergence must cease therefore at x = 1 , since the real axis leaves 
the circle of convergence at these points (see Fig. 111). 

Finally, as to the convergence of the series on the 
unit circle itself, I shall give you the reference which 
came up when we were talking about the connection 
between power series and trigonometric series. The 
'* 7 convergence depends upon whether or not the real and 
the imaginary part of the function, in view of the 
singularities that must exist on the circle of conver- 
gence, can be developed there into a convergent tri- 
gonometric series. 

I should like now to enliven the discussion of Taylor's theorem by 
showing its relations to the problems of interpolation and of finite 
differences. There, also, we are concerned with the approximation to 


Fig. 110. 

Fig. 111. 

Taylor's Theorem. 


Fig. 112. 

a given curve by means of a parabola. But instead of trying to make 
the parabola fit as closely as possible at one point, we require it to cut 
the given curve in a number of preassigned points; and the question 
is, again, as to how far this interpolation parabola gives a tolerable 
approximation. In the simplest case, this amounts to replacing the curve 
by a secant instead of the tangent (see Fig. 112). Similarly one passes 
a quadratic parabola through three points 
of the given curve, then a cubic parabola 
through four points, and so on. 

This is a natural way of approaching 
interpolation, one that is very often em- 
ployed, e.g., in the use of logarithmic 
tables There we assume that the logarithmic curve runs rectilinearly 
between two given tabular values and we interpolate "linearly" in the 
well known way, which is facilitated by the difference tables. If this 
approximation is not close enough, we apply quadratic interpolation. 

From this broad statement of the general problem, we get a deter- 
mination of the osculating parabolas in Taylor's theorem as a special 
case, that is, when we simply allow the intersections with the inter- 
polation parabolas to coincide. To be sure, the replacing of the curve 
by these osculating parabolas is not properly expressed by the word 
"interpolation" , except that one includes "extrapolation" in the problem 
of interpolation. For example, the curve is compared not only with 
the part of the secant lying between its points of intersection, but also 
with the part beyond. For the entire pro- 
cess the comprehensive word approximation 
seems more suitable. 

I shall now give the most important 
formulas of interpolation. Let us first de- 
termine the parabolas of order n \ which 

cut the given function in the points x = a lt a 2 , . . ., a n , that is, whose 
ordinates in these points are f(a^ } /(# 2 ), . . ., f(a n ) (see Fig. 113). This 
problem, as you know, is solved by Lagrange's interpolation formula 

(L f CU Z 

Fig. 113. 


y = 


(x - a z ) (x - a 3 ) 

- ^2) K - 8 ) 

(x aj) (x - a 3 ) 

- a x ) (a 2 - a 8 


It contains n terms with the factors / (a^ , / (0 2 ) ,, 
numerators lack in succession the factors (x a x ) , (x a 2 ) , . . . , 
(x a n ) . It is easy to verify the correctness of the formula. For, 
each summand of y, and hence y itself, is a polynomial in x of degree 
n \ . If we put x = #! all the fractions vanish except the first, which 

230 Analysis; Concerning Infinitesimal Calculus Proper. 

reduces to 1 , so that we get y = / (a^ , Similarly we get y = f (a 2 ) for 
% = 2 , etc. 

From this formula it is easy to derive, by specialization, one that is 
often called Newton's formula. This has to do with the case where 
the abscissas a lt . . ., a n are equidistant (see Fig. 114). As the notation 
* of the calculus of finite differences is advan- 
/ tageous here we shall first introduce it. 

Let Ax be any increment of x and let Af(x) 
be the corresponding increment of f(x) so that 

Now Af(x) is also a function of x which, if 

we change x by Ax, will have a definite difference called the second 
difference, A*f (x) t so that 

In the same way we have 

A*f(x + Ax) = A*f(x) + A*f(x) , etc. 

This notation is precisely analogous to that of differential calculus, 
except that one is concerned here with finite quantities and there is 
no passing to the limit. 

From the above definitions of differences there follows at once for 
the values of / at the successive equidistant places 

f(x+ Ax) =f(x) + Af(x), 

f(x + 2Ax) = f(x + Ax) + Af(x + Ax) 


f(x + 4Ax) = f(x) + 4Af(x) + 6A*f(x) + 4A*f(x) - 

This table could be continued, the values at equidistant points being 
expressed by means of successive differences taken at the initial point 
and involving the binomial coefficients as factors, 

Newton's formula for the interpolation parabola of order (n 1) 
belonging to the n equidistant points of the x axis, 

that is, which has at these points the same ordinates as / (x) , will be 

f/ a \ I v* i "i v") i \ **/ \" * ""i tLJ-LL I 
J / W i " -i | Ay. " o! (A*\* ' 


(x a) (x a A x) (x a (n 


This is, in fact, a polynomial in * of order n 1 . For x = a it reduces 

Taylor's Theorem. 231 

to / (a); for x = a + Ax all the terms, except the first two, become 
zero and there remains y == / (a) + Af (a), which by (2) is equal to 
/ (a + Ax); and so on. Thus the table (2) yields a polynomial which 
assumes the correct values at all the n places. 

If we wish to use this interpolation formula to real advantage, 
however, we must know something as to the correctness with which it 
represents /(#), that is, we must be able to estimate the remainder. 
Cauchy gave 1 the formula for this in 1840, and I should like to derive it. 
I shall start from the more general Lagrange formula. Let x be any 
value between the values a lf a 2 , . . ., a n , or beyond them (interpolation 
or extrapolation). We denote by P (x) the ordinate of the interpolation 
parabola given by the formula and by R (x) the remainder 

(4) /(*) = P (*) + *(*). 

According to the definition of P (x) the remainder R vanishes for 

x = a lf a 2 , . . . , a n and we therefore set 

R(x} = (jLz*)fr----(-*)y(a). 

It is convenient to take out the factor n \ Then it turns out, in complete 
analogy with the remainder term of Taylor's series, that \p (x) is equal 
to the n-th derivative of f (x) taken for a value x = lying between the 
n 1 points a lf a 2 , . . ., a n , x. This assertion that the deviation of / (x) 
from the polynomial of order n 1 depends upon the entire course 
of the function /< n ) (x) seems entirely plausible, if we reflect that / (x) 
is equal to that polynomial when f^ (x) vanishes. 

As to the proof of the remainder formula, we derive it by the following 
device. Let us set up, as a function of a new variable z t the expression 

where x remains as a parameter in v ; (x) . Now F (a^ = F (a 2 ) 
= F (a n ) = 0, since P (aj = / (aJ.P (a 2 ] = f (a,), . . ., P (a n ) = f (a n ) 
by definition. Furthermore F (x) = because the last summand goes 
over into R (x), for z = x, so that the right side vanishes by (4). We 
know, therefore, n + 1 zeros z = a lt a 2 , . . ., a n , x, of F(z). Now 
apply ttie extended meap-value theorem, which one gets by repeated 
application of the ordinary theorem (p. 213), namely: // a continuous 
function, together with its first n derivatives, vanishes at n + 1 points, 
then the n-th derivative vanishes at one point, at least, which lies in the 
interval containing all the zeros. Hence if / (z) , and therefore also F (z) , 
has n continuous derivatives, there must be a value f between the 
extremes of the values a l , a 2 , . . ; , , x for which 


1 Comptes Rendus, vol. 11, pp. 775 789- CEuvres, 1st series, vol. 5, pp, 409 
to 424, Paris, 1885- 

232 Analysis: Concerning Infinitesimal Calculus Proper. 

But we have 


since the polynomial P (z) of degree n 1 has for its n-ih derivative 
and since only z n if (x)/n\ f the highest term of the last summand, has 
an n-th derivative which does not vanish. Therefore we have, finally 

vw = > r v.w : 

which we wished to prove. 

I shall write down Newton's interpolation formula with its remainder 


(x a) (x a Ax) 


-jT- -37- 2! 

(AT- a) \x-a- (n-2)Ax] 
+ " (n- 1)! 

(*-fl) [*-0- (n- 


where f is a mean value in the interval containing the n 1 points a . 
a + Ax, a + 2 <4tf , ..., + ( !) ^#, #. The formula (5) is, in fact, 
indispensable in the applications. I have already alluded to linear inter- 
polation when logarithmic tables are used. If / (x) = log x and n = 2, 
we find, from (5) 

, , . x a Aloga (x a) (x a Ax) M 

log* = loga + _ -- - [ - -- ^ ---- i - ? . 

Since d* log x/dx^ = Mix 2 where M is the modulus of the logarithmic 
system. Hence we have an expression for the error which we commit 
when we interpolate linearly between the tabular logarithms for a and 
a + A x . This error has different signs according as x lies between a 
and a + Ax or outside this interval. Every one who has to do with 
logarithmic tables should really know this formula. 

I shall not devote any more attention to applications, but shall now 
draw your attention to the marked analogy between the interpolation 
formula of Newton and the formula of Taylor. There is a substantial 
reason for this analogy. It is easy to give an exact deduction of Taylor's 
theorem from the Newtonian formula, corresponding to the passage to 
the limit from interpolation parabolas to osculating parabolas. Thus, 
if we keep x, a, and n fixed and let A% converge to zero, then, since 
/ (x) has n derivatives, the n 1 difference quotients in (5) go over 
into the derivatives 

Af(a) ,,, x ,. A 2 f(a) //// \ 

~ = - = 

In the last term of (5), the value of f can change with decreasing Ax. 
Since all the other terms on the right have definite limits, however, 
and the left side has the fixed value / (x) during the entire limit process, 
it follows that the values of / (n ^(l) must converge to a definite value 

Taylor's Theorem. 233 

and that this value, furthermore, must, because of the continuity of / (n) , 
be a value of this function for some place between a and x . If we denote 
this again by I we have 

/ W = /(*) + i=f (a) + - - - + ? /*-() + = 

Thus we have obtained a complete proof of Taylor's theorem with the 
remainder term and at the same time have given it an ordered place 
in the theory of interpolation. 

It seems to me that this proof of Taylor's theorem, which brings 
it into wider relation with very simple questions and which provides 
such a smooth passage to the limit, is the very best possible one. But 
all the mathematicians to whom these things are familiar (it is remark- 
able that they are unknown to many, including perhaps even some 
writers of textbooks) do not think so. They are accustomed to confront 
a passage to a limit with a very grave face and would therefore prefer 
a direct proof of Taylor's theorem to one linking it with the calculus 
of finite differences. 

I must emphasize however that, as a matter of history, the source 
of Taylor's theorem is actually the calculus of finite differences. 
I have already mentioned that Brook Taylor first published it in his 
Methodus incrementorum 1 . He first deduces Newton's formula, without 
the remainder, of course, and then puts in it Ax = and n = oo. He 
thus gets correctly from first terms of Newton's formula the first terms 
of his new series: 

The continuation of this series, according to the same law, seems to him 
self evident, and he gives no thought either to a remainder term or to 
convergence. We have here, in fact, a passage to the limit of unexampled 
audacity. The earlier terms, in which x a Ax, x a 2Ax, ... 
appear, offer no difficulty, because these finite multiples of A x approach 
zero with A x ; but with increasing n there appear terms in ever increasing 
number, ^presenting factors x a kAx with larger and larger k, and 
.one is not justified in treating these forthwith in the same way and in 
assuming that they go over into a convergent series. 

Taylor really operates here with infinitely small quantities (differen- 
tials) in the same unquestioning way as the Leibnizians. It is interesting 
to reflect that although, as a young man of twenty-nine, he was under 
the eye of Newton, he departed from the latter's method of limits. 

You will find an excellent critical presentation of the entire develop- 
ment of Taylor's theorem in Alfred Pringsheim's memoir: Zur Geschichte 

1 Londini, 1715, p. 21-23- 

234 Analysis: Concerning Infinitesimal Calculus Proper. 

des Taylorschen Lehrsatzes 1 . I should like to speak here about the 
customary distinction between Taylor's series and that of Maclaurin. 
As is well known, many textbooks make a point of putting a = and 
of calling the obvious special case of Taylor's series which thus arises: 

/w = /(o) + ^f(o) + ~no)-+--- 

the series of Maclaurin ; and many persons may think that this distinction 
is important. Anybody who understands the situation however sees 
that it is comparitively unimportant mathematically. But it is not 
so well known that, considered historically, it is pure nonsense. For 
Taylor had undoubted priority with his general theorem, deduced in 
the way indicated above. More than this, he emphasizes at a later 
place in his book (p. 27) the special form of the series f or a and 
remarks that it could be derived directly by the method which is called 
today that of undetermined coefficients. Furthermore, Maclaurin took 
over 2 this deduction in 1742 in his Treatise of Fluxions (which we 
mentioned on p. 212) where he quoted Taylor expressly and made no 
claim whatever of offering anything new. But the quotation seems to 
have been disregarded and the author of the book seems to have been 
looked upon as the discoverer of the theorem. Errors of this sort are 
common. It was only later that people went back to Taylor and named 
the general theorem, at least, after him. It is difficult, if not impossible, 
to overcome such deeprooted absurdities. At best, one can only spread 
the truth in the small circle of those who have historical interests. 
I shall now supplement our discussion of infinitesimal calculus with 
some remarks of a general nature. 

3. Historical and Pedagogical Considerations 

I should like to remind you, first of all, that the bond which Taylor 
established between difference calculus and differential calculus held for 
a long time. These two branches always went hand in hand, still in the 
analytic developments of Euler, and the formulas of differential calculus 
appeared as limiting cases of elementary relations that occur in the 
difference calculus. This natural connection was first brokeh by the 
oft mentioned formal definitions of Lagrange's derivative calculus'. 
I should like to show you a compilation from the end of the eighteenth 
century which, closely following Lagrange, brings together all the facts 
then known about infinitesimal calculus, namely the Traite du Calcul 
Differentiel et du Calcul Integral of Lacroix 3 . As a characteristic sample 
from this work, consider the definition of the derivative (vol. I, p. 145)'- 

1 Bibliotheca Mathematica, 3rd series, voL I (1900), p. 433 479- 

2 Edinburgh, 1742, vol.11, p. 610. 

3 Three volumes, Paris, 17971800, with many later editions. 

Historical and Pedagogical Considerations. 2)5 

A function / (x) is defined by means of a power series. By using the 
binomial theorem (and rearranging the terms) one has 

Lacroix now denotes the term of this series which is linear in h by df (x) , 
and, writing dx for h itself, he has for the derivative, which he calls diffe- 
rential coefficient 

Thus this formula is deduced in a manner thoroughly superficial even 
if unassailable. Within the range of these thoughts, Lacroix could not, 
of course, use the calculus of differences as a starting point. However, 
since this branch seemed to him too important in practice to be omitted, 
he adopted the expedient of developing it independently, which he did 
very thoroughly in a third volume, but without any connecting bridge 
between it* and differential calculus. 

This "large Lacroix" is historically significant as the proper source 
of the many textbooks of infinitesimal calculus which appeared in the 
nineteenth century. In the first rank of these I should mention his 
own textbook, the "small Lacroix" 1 . 

Since the twenties of the last century the textbooks have been 
strongly influenced also by the method of limits which Cauchy raised 
to such an honorable place. Here we should first think of the many 
French textbooks, most of which, as Cours d' Analyse de VEcole Poly- 
technique, were prepared expressly for university instruction. Directly 
or indirectly, German textbooks also have depended on them, with the 
single exception, perhaps, of the one by Schlomilch. From the long 
list of books, I shall single out only Serret's Cours deCalcul Difftrentiel 
et Integral, which appeared first in 1869 in Paris. It was translated into 
German in 1884 by Axel Harnack and has been since then one of our 
most widely used textbooks. It suffered as to symmetry at the hands 
of a long series of revisers. The editions 2 which have appeared since 
1906, however, have been subjected to a thoroughgoing revision by 
G. Scheffers of Charlottenburg, the result being a homogeneous work. 
I am glad to mention also an entirely new French book, the Cours 
d' Analyse Mathematique by Goursat 3 in three volumes, which is fuller 
in many ways than Serret and contains, in particular, a long series of 
entirely modern developments. Furthermore it is a very readable book. 

1 TraM Ettmentaire du Calcul Difffrentiel et Integral. Two volumes, Paris, 1797. 

2 Since 1906: Serret, J, A., u. G. Scheffers, Lehrbuch der Differential- und 
Integralrechnung, vol. I, sixth edition. Leipzig 1915; vol. II, 67 edition; vol. Ill, 
fifth edition, 1914. 

8 Paris 19021907, vol. I, third edition. 1917; vol. II, third edition. 1918; 
vol. Ill, second edition. 1915- (Translated into English: vol. I by E. R. Hedrick, 
1904, Ginn and Co.; vol. II by E. R. Hedrick and O. Dunkel, 1916, Ginn and Co.) 

Analysis: Concerning Infinitesimal Calculus Proper. 

In all these recent books, the derivative and the integral are based 
entirely upon the concept of limit. There is never any question as to 
difference calculus or interpolation. One sees the thing in a clearer 
light, perhaps, in this way, but, on the other hand, the field of view is 
considerably narrowed, as it is when we use a microscope. Difference 
calculus is now left entirely to the practical calculators, who are obliged 
to use it, especially the astronomers; and the mathematician hears 
nothing of it. We may hope that the future will bring a change 1 here. 

As a conclusion of my discussion of infinitesimal calculus I should 
like to bring up again for emphasis four points, in which my exposition 
differs especially from the customary presentation in the textbooks: 

1. Illustration of abstract considerations by means of figures (curves 
of approximation, in the case of Fourier's and Taylor's series). 

2. Emphasis upon its relation to neighboring fields, such as calculus 
of differences and of interpolation, and finally to philosophical investiga- 

3. Emphasis upon historical growth. 

4. Exhibition of samples of popular literature to mark the difference 
between the notions of the public, as influenced by this literature and those 
of the trained mathematician. 

It seems to me extremely important that precisely the prospective 
teacher should take account of all of these. As soon as you begin teaching 
you will be confronted with the popular views. If you lack orientation, 
if you are not well informed concerning the intuitive elements of mathe- 
matics as well as the vital relations with neighboring fields, if, above 
all, you do not know the historical development, your footing will be 
very insecure. You will then either withdraw to the ground of the 
most modern pure mathematics, and fail to be understood in the school, 
or you will succumb to the assault, give up what you learned at the 
university, and even in your teaching allow yourself to be buried in 
the traditional routine. The discontinuity between school and uni- 
versity, of which I have often spoken, is greatest precisely in the field 
of infinitesimal calculus. I hope that my words may contribute to its 
removal and that they may provide you with useful armor in your 

This brings me to the end of the conventional analysis. By way of 
supplement I shall discuss a few theories of modern mathematics to 
which I have referred occasionally and with which I think the teacher 
should have some acquaintance. 

1 In order to make a beginning here, I induced Friesendorff and Prumm to 
translate Markoffs Differenzenrechnung into German (Leipzig, 1896). There is 
a series of articles in the Enzyklopadie. A work on Differenzenrechnung by E. Nor- 
lund has just appeared (Berlin, Julius Springer, 1924) which exhibits the subject 
in new light. 


I. Transcendence of the Numbers e and a 

The first topic which I shall discuss will be the numbers e and n. 
In particular, I wish to prove that they are transcendental numbers. 

Interest in the number n, in geometric form, dates from ancient 
times. Even then it was usual to distinguish between the problem of 
its approximate calculation and that of its exact theoretical construction; 
and one had certain fundamentals for the solution of both problems. 
Archimedes made an essential advance, in the first, with his process of 
approximating to the circle by means of inscribed and circumscribed 
polygons. The second problem soon centered in the question as to 
whether or not it was possible to construct n with ruler and compasses. 
This was attempted in all possible ways with never a suspicion that the 
reason for continued failure was the impossibility of the construction. 
An account of some of the early attempts has been published by Rudio 1 . 
The quadrature of the circle still remains one of the most popular 
problems, and many persons, as I have already remarked, seek salvation 
in its solution, without knowing, or believing, that modern science has 
long since settled the question. 

In fact, these ancient problems are completely solved today. One 
is sometimes inclined to doubt whether human knowledge really can 
advance, and in some fields the doubt may be justified. In mathematics, 
however, there are indeed advances of which we have here an example. 

The foundations upon which the modern solution of these problems 
rests date from the period between Newton and Euler. A valuable tool 
for the approximate calculation of n was supplied by infinite series, a 
tool whith made possible an accuracy adequate for all needs. The 
most elaborate result obtained was that of the Englishman Shanks, who 
calculated n to 707 places 2 . One can ascribe this feat to a sportsmanlike 
interest in making a record, since no applications -could ever require 
such accuracy. 

On the theoretical side, we find the number e, the base of the system 
of natural logarithms, coming into the investigations during the same 

1 Der Bericht des Simplicius uber die Quadraturen des Antiphon und Hippokrates. 
Leipzig, 1908. 

2 See Weber- Wellstein, vol. l, p. 523- 

2^8 Supplement: Transcendence of the Numbers e and n. 

period. The remarkable relation e in = 1 was discovered and a means 
was developed in the integral calculus which, as we shall see, was of 
importance for the final solution of the question as to the quadrature 
of the circle. The decisive step in the solution of the problem was taken 
by Hermite 1 in 1873, when he proved the transcendence of e. He did 
not succeed in proving the transcendence of n. That was done by 
Lindemann 2 in 1882. 

These results represent an essential generalization of the classical 
problem. That was concerned only with the construction of n by means 
of ruler and compasses, which amounts, analytically, as we saw (p. 51) 
to representing n by a finite succession of square roots and rational 
numbers. But the modern results prove not merely the impossibility 
of this representation; they show far more, namely, that n (and like- 
wise e) is transcendental, that is, that it satisfies no algebraic relation 
whatever whose coefficients are integers. In other words, neither e 
nor n can be the root of an algebraic equation with integral coefficients : 

a Q + ax + a 2 x 2 + - - + a n x n = 

no matter how large the integers a Q> . . ., a n or the degree n. It is 
essential that the coefficients be integers. It would suffice however to 
say fractions, since we could make them integral by multiplying through 
by a common denominator. 

I pass now to the proof of the transcendence of e, in which I shall 
follow the simplified method given by Hilbert in Volume 43 of the 
Mathematische Annalen (1893). We shall show that the assumption 
of an equation 

(1) a + a^e + a 2 e 2 + - - - + a n e n = 0, where a -+ 0, 

in which , . . .,a n are integers, leads to a contradiction. This will 
involve the use of only the simplest properties of whole numbers. We 
shall need, namely, from the theory of numbers, only the most ele- 
mentary theorems on divisibility, in particular, that an integer can be 
separated into prime factors in only one way, and, second, that the 
number of primes is infinite. 

The plan of the proof is as follows. We shall set up a method which 
enables one to approximate especially well to e and powers of e, by. 
meajis of rational numbers, so that we have 

M * _ M l+ g l ,2 _ M 2 + *2 _ W _ M n + *V 

e - ' e - ' ' ' " e ~ ~~ 

where M , M lt M 2 , . . ., M n are integers, and t /M, e 2 /M, . . ., n /M are 

1 Comptes Rendus, vol. 77 (1873), p. 18-24, 74-79, 226-233, 285-293; 
== Werke III (1912), p. 150. 

2 Sitzungsberichte der Berliner Akademie, 1882, p. 679. and Mathematische 
Annalen, vol.20 (1882), p. 213- 

Transcendence of e. 


very small positive fractions. Then the assumed equation (1), after 
multiplication by M, takes the form 

(3) [a Q M+a l M l + a^M 2 -\ ----- h *MJ + \a^+ a^-\ ----- Va n e n } =0. 

The first parenthesis is an integer, and we shall prove that it is not 
zero. As for the second parenthesis, we shall show that e lt . . ., e n can 
be made so small that it will be a positive proper fraction. Then we 
shall have the obvious contradiction that an integer a Q M + a l M l + 
+ a n M n which is not zero, increased by a proper fraction a^ + 
+ a n e n is zero. This will show the impossibility of (1). 

In the course of the discussion which I have just outlined we shall 
make use of the theorem that if an integer is not divisible by a definite 
number, the integer cannot be zero (for zero is divisible by every number). 
We shall show, namely, that M x , . . . , M n are divisible by a certain 
prime number p, but that a Q M does not contain p, and that, therefore, 
a Q M + a l M l + + a n M n is not divisible by p, and hence is different 
from zero. 

The principal aid in carrying out the indicated proof comes from 
a certain definite integral which was devised by Hermite for this pur- 
pose and which we shall call Hermite' s integral. The key to this proof 
lies in its structure. This integral, whose value, as we shall see, is a 
positive whole number and which we shall use to define M , is 

~ -. 

(P l) ! 

where n is the degree of the assumed equation (1), and p is an odd 
prime which we shall determine later. From this integral we shall get 
the desired approximation (2) to the powers e v (v = 1 , 2 , . . . , n) by 
breaking the interval of integration of the integral M e v at the point v 
and setting 

Uti M - * r* P ~ l[( *~ 1) "- ('-*)]'*"' dz 

(4a) Mv ~ e } v (f=W ' 

/,u\ v - ... - . 

(4b) , e v = e' ______ dz . 

Let us now take up the details of the proof. 
1. We start with the well known formula from the beginnings of 
the theory of the gamma function: 


We shall need this formula only for integral values of Q , in which case 
= (Q 1)!, and I shall deduce it under this restriction. If we 


Supplement: Transcendence of the Numbers e and a. 

integrate by parts we have, for Q > 1 : 


= (Q i)fze- 2 e- z dz. 

The integral on the right is of the same form as the one on the left, 
except that the exponent of z is reduced. If we apply this process 
repeatedly we must eventually come to 2, since Q is an integer; and 


since / e~ z dz = 1 , we obtain finally 

(5) r#-* 


= ( e - i) ( e - 2) . . . 3 2 i = (e - 1) i 

Thus for integral Q the integral is a whole number which increases 
very rapidly when Q increases. 

To make this result clear geometrically, let us draw the curve 
y = z Q ~ l e~ z for different values of Q. The value of the integral will 
then be represented by the area under the curve extending to infinity 

(see Fig. 115). The larger 

* Q is the more closely the 

curve hugs the z axis at 
the origin, but the more 
rapidly it rises beyond 
z = 1 . The curve has a 
maximum at z = Q 1 , 
for all values of g; in 
other words the maximum 
occurs farther and farther 

to the right as Q increases ; and its value also increases with Q . To the 
right of the maximum, the factor e~ z prevails so that the curve falls, 
approaching the z axis asymptotically. It is thus comprehensible that 
the area (our' integral) always remains finite but increases rapidly with Q. 
2. With this formula we can now easily evalute our Hermite integral. 
Developing the integrand by the polynomial theorem 

1 z 3 14 5 


Fig. 115. 

where only the terms involving the highest and the lowest powers of z 
have been written down, the integral becomes 



Transcendence of e. 241 

The C Q are integral constants, by the polynomial theorem. Now we 
can apply formula (5) to each of these integrals and obtain 


The summation index Q is always larger than p and consequently 
(Q !)!/( 1)! is an integer and one which contains p as a factor, 
so that we can take p as a factor out of the entire sum: 


Now, so far as divisibility by p is concerned, M must behave like 
the first summand ( l) n (n!) p . And since p is a prime number it will 
not be a divisor of this summand if it is not a divisor of any of its factors 
1 , 2 , . . . , n, which will certainly be the case if p > n . But this condition 
can be satisfied in an unlimited number of ways, since the number of 
primes is infinite. Consequently we can bring it about that ( \) n (n\) p , 
and hence M, is not divisible by p. 

Since furthermore a Q =)= 0, we can see to it, at the same time, that 
fl is not divisible by p by selecting p larger also than |a |, which is, 
of course, possible, by what was said above. But then the product 
M will not be divisible by p , and that is what we wished to show. 

3. Now we must examine the numbers M v (v = \ , 2, . . . , n) , defined 
in (4 a) (p. 239). Putting the factor e v under the sign of integration and 
introducing the new variable of integration f = z v , which varies 
from to oo when z runs from v to oo, we have 

_ r 


This expression has a form entirely analogous to the one considered 
before for M and we can treat it in the same way. If we multiply out 
the factors of the integrand there will result an aggregate of powers 
with integral coefficients of which the lowest will be f. The integral 
of the numerator will thus be a combination of the integrals 

fFc-ed, rV^-^f, ..., r 
Jo Jo Jo 

and since these are, by (5), equal to p\,(p + 1)!, ... the numerator 
will be p I multiplied by a whole number A , so that we have 

M * = - =#-4,, (" = 1, 2, . . . , n) . 

In other words, every M v is a whole number which is divisible by p. 
This, combined with the result of No. 2, proves the statement made 
on p. 239 that a M + a l M l + + a n M n is not divisible by p and 
is therefore different from zero. 

242 Supplement: Transcendence of the Numbers e and n. 

4. The second part of the proof has to do with the sum a^ s l + 
+ a n e n , where, by (4b), 

" 1 [(^- l)(*-2) ...(*- 

v ~jo 

We must show that these e v can be made arbitrarily small by an appro- 
priate choice of p . To this end we use the fact that we can make p as 
large as we chose; for the only conditions thus far imposed upon p are 
that it should be a prime number larger than n and also larger than 
|0 |, and these can be satisfied by arbitrarily large prime numbers. 
Let us examine the graph of the integrand. At z = it will be 
tangent to the z axis, but at z = 1 , 2, . . . , n (in Fig. 116, n = 3) it 

will be tangent to the 
z axis and also cut it, 
since p is odd. As we 
shall see soon, the 
presence in the deno- 
minator of (p 1)! 
brings it about that 
for large p the curve 
y ' * * * departs but little 

Fig. 116. , from the z axis in the 

interval (0, n), so 

that it seems plausible that the integrals s v should be very small. 
For z > n the curve rises and runs asymptotically like the former 
curve z e ~ l e~ z [iQi Q = (n + \)p] and finally approaches the z axis. It 
was for this reason that the value M of the integral (when the interval 
of integration was from to oo) increased so rapidly with p. 

In actually estimating the integrals we can be satisfied with a rough 
approximation. Let G and g v be the maxima of the absolute values 
of the functions z (z 1) ... (z n) and (z 1) (z 2) ... (z n) e~ z+v 
respectively in the interval (Q,ri): 

Since the integral of a function is never larger than the integral of its 
absolute value, we have, for each E V 


Now G, g v , and v are fixed numbers independent of p, but the number 
(/> !)! in the denominator increases ultimately more rapidly than 
the power G p ~\ or, more exactly, the fraction G p ~ l /(p 1)! becomes, 
for sufficiently large p, smaller than any preassigned number, however 
small. Thus, because of (6), we can actually make each of the n numbers s v 
arbitrarily small by choosing p sufficiently large. 

Transcendence of a. f ' 243 

It follows immediately from this that we can also make the sum of n 
terms a e l + + a n e n arbitrarily small. We have, in fact 

and by (6) 

/i i j i i i /* i .it \ G p-1 

^(lir i'fo + li|-2gi+ + W-w-gn) -77 n 

Since the parenthesis has a value which is independent of p, we can, 
by virtue of the factor G p ~ l /(p 1)!, make the entire right hand side, 
and hence also a e + a 2 e 2 + + #n n, as small as we choose, and, 
in particular, smaller than unity. 

With this we have shown, as we agreed to do (p. 239), that the. 
assumption of the equation (3) 

leads to a contradiction, namely that a non vanishing integer increased 
by a proper fraction gives zero. And since this equation cannot exist 
the transcendence of e is proved. 

Proof of the Transcendence of n 

We turn now to the proof of the transcendence of the number n. 
This proof is somewhat more difficult than the foregoing, but it is still 
fairly easy. It is only necessary to begin at the right end, which is 
indeed the art of all mathematical discovery. ' 

The problem, as Lindemann considered it, was the following: It has 


been shown thus far that an equation ^?a v e v = cannot exist if the 

coefficients a v and the exponents v of e are ordinary whole numbers. 
Would it not be possible to prove a similar thing where a v and v are 
arbitrary algebraic numbers? He succeeded in doing this; in fact, his most 
general theorem concerning the exponential function is as follows: An 


equation^a v e b v = cannot exist if the a v , b v are algebraic numbers, whereby 

the a v are arbitrary, the b v different from one another. The transcendence 
of n is then a corollary to this theorem. For, as is well known, 1 + & in ; 
and if n were an algebraic number, i n would be also, and the existence 
of this equation would contradict the above theorem of Lindemann. 
I shall now prove in detail only a certain special case of Linde- 
mann's theorem, one which carries with it, however, the transcendence 
of n. I shall follow again, in the main, Hilbert's proof in Volume 43 of 
the Mathematische Annalen, which is essentially simpler than Linde- 
mann' s, and which is an exact generalization of the discussion which 
we have given for e. 

244 Supplement: Transcendence of the Numbers e and a. 

The starting point is the relation 
(1) 1 + e in = 0. 

If, now, n satisfies any algebraic equation with integral coefficients 
then in also satisfies such an equation. Let a l9 <x 2 , . . ., <x n be all the 
roots, including i n itself, of this last equation. Then we must also 
have, because of (1): 

(1 + e**)(\ +^ a ) (1 + *") = (). 
Multiplying out we obtain 

e 01 ^"* -{- - + e**~~ l+ **) 


1 ' * -j- (!+*+ +**) = 0. 

Now some of the exponents which appear here might, by chance, be 
zero. Everytime that this occurs the left hand sum has a positive 
summand 1 , and we combine these, together with the 1 that appears 
formally, into a positive integer , which is certainly different from 
zero. The remaining exponents, all different from zero, we denote by 
Pi> @2> > PN and we write, accordingly, instead of (2) , 

(3) a v + ^ + ^ + - + eP N = 0, where > . 

Now Pi, . . ., (} N are the roots of an algebraic equation with integral 
coefficients. For, from the equation whose roots are a x , . . ., oc n we 
can construct one of the same character whose roots are the two term 
sums x + 2 a i + a s t* 1611 another for the three term sums 

#1 + ^2 + a a> a i + ^2 + *4f an d so on; finally, 04 + <* 2 H 1- a n 

is itself rational and satisfies therefore a linear integral equation. By 
multiplying together all these equations, we obtain again an equation 
with integral coefficients, which might have some zero roots, and whose 
remaining roots are /8 1 , . . ., fi N . Omitting the power of the unknown 
which corresponds to the zero roots there will remain for the N quanti- 
ties /8 an algebraic equation of degree N with integral coefficients and 
absolute term different from zero 

(4) b + b^z + b z z* + + b N z N = 0, where 6 0f b N =j= 0. 

We now have to prove the following special case of Liqdemann's 
theorem. An equation of the form (3), with integral non-vanishing 0t , 
cannot exist if fi^, . . . , (} N are the roots of an algebraic equation of degree N, 
with integral coefficients. This theorem includes the transcendence of n. 

The proof involves the same steps as the one already given for the 
transcendence of e. Just as we could there approximate closely to the 
powers e 1 , e 2 , . . ., e n by means of rational numbers, so we shall be 
concerned here with the best possible approximation to the powers 
of e which appear in (3), and we shall write., in the old notation, 

lt\ tfi M I + S I ^ _ M Z + 
W * - M ' * ~ M 

Transcendence of si. 245 

where the denominator M is again an ordinary integer but M l , . . . , M N 
are not integers as formerly, but are integral algebraic numbers, and 
the /?!, . . ., fa, which in general can now be complex, are in absolute 
value very small. It is here that the difficulty in this proof lies, as 
compared to the earlier one. The sum of all the Af lf . . . , M N will again, 
however, be ah integer, and we shall be able to arrange it so that the 
first summand in the equation: 

(6) [a M + M l + M z + + My] + [fi x + f. + + *N\ = 0, 

[into which (3) goes over when we multiply by M and use (5)] will be 
a non-vanishing integer, while the second summand will be, in absolute 
value, smaller than unity. Essentially, this will be the same contradiction 
which we used before. It will show the impossibility of (6) and (3) 
and so complete our proof. As to detail, we shall again show that M l 
+ M 2 + + M N is divisible by a certain prime number p , but that 
a Q M is not, which will show that the first summand in (6) cannot 
vanish; then we shall choose p so large that the second summand will 
be arbitrarily small. 

1. Our first concern is to define M by a suitable generalization of 
Hermite's integral. A hint here lies in the fact that the zeros of the 
factor (z 1) (z 2) ... (z n) in Hermite's integral were the ex- 
ponents of e in the hypothetical algebraic equation. Hence we now 
replace that factor by the product made by using the exponents in (3), 
i.e., the solutions in (4): 

(7) (z - A)(* - A) ...(*-&)= [6 + M + 

It turns out to be essential here to put in a suitable power of b N as 
factor, which was unnecessary before because (z 1) ... (z n) was 
integral. We set then finally 

(8) M 

2. Just as before, we now develop the integrand of M according to 
powers of z. The lowest power, that belonging to z p ~ l , gives then: 

where the integral has been evaluated by means of the gamma formula 
(p. 239). The remaining summands in the integrand contain either z p 
or still higher powers, so that the integrals contain the factor p\/p 1) ! , 
multiplied by integers, and are thus all divisible by p . Consequently M 
is an integer which is certainly not divisible by p, i.e., provided the 
prime number p is not a divisor of either 6 or b$ . But since these two 
numbers are both different from zero, we can bring this about by 
choosing p so that p > |6 | and also p > \b N \. 

246 Supplement: Transcendence of the Numbers e and n. 

Since a > it follows that a Q M is not divisible by p if we impose 
the additional condition p > . Inasmuch as the number of primes 
is infinite we can satisfy all these conditions in an unlimited number 
of ways. 

3. We must now set up M v and e v . Here we must modify our 
earlier plan because the p v , which now take the place of the old v , can 
be complex; in fact one them is in. If we are to split the integral M 
as we did before we must first determine the path of integration in the 
complex plane. Fortunately the integrand of our integral is a finite 
single-valued function of the variable of integration z, regular every- 
where except at z = oo, where it has an essential singularity. Instead 
of integrating from to oo along the real axis we can choose any other 
path from tooo, provided it ultimately runs asymptotically parallel 
/\ to the positive real half axis. This is 

2-Plan6 /' \ necessary if the integral is to have a 

meaning at all, in view of the behavior 
of e~ z in the complex plane. 

Let us now mark the N points ft l9 
Pz, > , PN in the plane and recall 
that we shall obtain the same value 

2 for M if we first integrate rectilinearly 

from to one of the points fa and 
Fi then to oo along a parallel to the real 

axis (see Fig. 117). Along this path 
we can separate M into the two characteristic parts: The rectilinear 
path from to fa supplies the e v which will become arbitrarily small 
with increasing p; the parallel from fa to oo will supply the integral 
algebraic number My: 

(8a) , = > -~ [6. + b,z + 

(v = 1, 2, . . . , N) , 

//oo z p-l J 
" ' - Jr. V-i)V [6 + * * + + b zN} * b ~ 1]p ~ l 

These assumptions satisfy (5). Our choice of a rectilinear path of 
integration was made solely for convenience; a curvilinear path from 
to ft v would, of course yield the same value for e v , but it is easier to 
estimate the integral when the path is straight. Similarly, we might 
choose, instead of the horizontal path from fi v to oo, an arbitrary curve 
provided only that it approached the horizontal asymptotically; but 
that would be unnecessarily inconvenient. 

4. I will discuss first the estimation of the e v , because this involves 
nothing new if we only recall that the absolute value of a complex 
integral cannot be larger than the maximum of the absolute value of 
the integrand, multiplied by the length of the path of integration, 

Transcendence of n. 247 

.which, in our case, is | f{ v \ . The upper limit for e v would be, then, the 
product of G p ~ li l(p 1) ! by factors which are independent of p t where G 
denotes the maximum of \z(b$ + bz + + b N z^) b%~ l \ in a region 
which contains all the segments joining with the ft v . From this one 
may infer, as we did before, (p. 243), that, by sufficiently increasing p, 
the value of each e v and, therefore, the value of e l + + S N can be 
made as small as we please and, in particular, smaller than unity. 

5. It is only in the discussion of the M v that essentially new con- 
siderations enter, and these are, to be sure, only generalizations of our 
former reasoning, due to the fact that integral algebraic numbers take 
the place now of what were then integral rational numbers. We shall 
consider, as a whole, the sum: 

If we make use of (7) (p. 245) and replace, in each summand of the 
above summation, the polynomial in z by the product of the factors 
( z ~ Pi) '" ( z &0 an d introduce the new variable of integration 
C = z /?, which will run through real values from to oo, we obtain 

N N 


(9) 1-1 

/*00 g_^ 

which may be written = / ,. __ >, p $ (t) , 

where we use the abbreviation 


This sum ^P(f), like each of its N summands, is a polynomial in f . 
In each of the summands, one of the N quantities jS lt . . ., f} N plays a 
marked role; but if we consider the polynomial in f obtained by multi- 
plying out in $(?), we see that these N quantities appear, without 
preference, in the coefficients of the different powers of . In other 
words, each of these coefficients is a symmetric function of ^ , . . . , fa . 
'The multiplying out of the individual factors by the multinomial theorem 
permits the further inference that these functions /? x , . . . , fa are rational 
integral functions with rational integral coefficients. But according to 
a well known theorem in algebra, any rational symmetric function, with 
rational coefficients, of all the roots of a rational equation is itself 
always a rational number; and since the &, . . . 9 fa are all the roots 
of equation (4), the coefficients of #() are actually rational numbers. 

But, ipore than this, we need rational integral numbers. These are 
supplied by the power of by which occurs as a factor of (f). We can, 

248 Supplement: Transcendence of the Numbers e and a. 

in fact, distribute this power among all the linear factors which occur 
there and write 


In analogy with what we had earlier, the coefficients of f , when this 
polynomial is calculated, are rational integral symmetric functions of 
the products by Pi, b N j3 2 , . . ., b N fi N , with rational integral coefficients. 
But these N products are roots of the equation into which (4) goes if 
we replace z by z/by: 

If we multiply through by b% 1 this equation goes over into: 

(10) W'+ &i &$-** + + by-tbyZ*'* + by.iZ*-* + Z N = 0, 

that is, an equation with integral coefficients when the coefficient of 
the highest power is unity. Numbers which satisfy such an equation 
are called integral algebraic numbers, and we have the following refinement 
of the theorem mentioned above : Rational integral symmetric functions, 
with rational integral coefficients, of all the roots of an integral equation 
whose highest coefficient is unity (i.e., of integral algebraic numbers) are 
themselves rational integral numbers. You will find this theorem in text- 
books on algebra; and if it is not always enunciated in this precise 
form you can, nevertheless, by following the proof, convince yourselves 
of its correctness. 

Now the coefficients of the polynomial <&() actually satisfied the 
assumptions of this theorem so that they are rational integral numbers 
which we shall denote by A Q , A lt . . ., Ay p -i. We have, then, 

With this we have essentially reached our goal. For, if we carry 
out the integrations in the numerator, using our gamma formula (p. 239) > 
we obtain factors p\, (p + 1) I, (p + 2)1 . . ., since each term contains 
as factor a power of pot degree p or higher; and after division by (p 1) ! 
there remains everywhere as factor a multiple of p, while the other 


factors are rational integral numbers (the A Q , AI, A 2 , . . .). Thus ^M v 
is certainly a rational integral number divisible by p. re=1 

We saw (p. 246) that a M was not divisible by p , so that 


Transcendence of n. 249 

is necessarily a rational integral number which is not divisible by p and 
hence, in particular, different from zero. Therefore the equation (6) : 


cannot exist, for a non vanishing integer added to ^?e v , which was 

shown in No. 4 (p. 247) to be smaller than unity in absolute value, 
cannot yield zero. But this proves the special case of Lindemann's 
theorem which we enunciated above (p. 244) and which carries with it 
the transcendence of n. 

I should like to mention here another interesting special case of 
the general Lindemann theorem, namely, that in the equation 0^ = b 
the numbers 6, /? cannot both be algebraic, with the trivial exception 
/? = 0, b = 1 . In other words, the exponential function of an algebraic 
argument (i as well as the natural logarithm of an algebraic number b 
is, with this one exception, transcendental. This statement includes the 
transcendence of both e and n , the former for /? = 1 , the latter for 
b = \ (because e i71 = 1) . The proof of this theorem can be effected 
by an exact generalization of the last discussion. One would start 
from b eP instead of from 1 + e" as before. It would be necessary 
to take into account not only all the roots of the algebraic equation 
for {{, but also all the roots of the equation for b, in order to arrive 
at an equation analogous to (3), so that one would need more notation 
and the proof would be apparently less perspicuous; but it would require 
no essentially new ideas. 

I shall not go farther into these proofs, but I should like to point 
out graphically the significance of the last theorem concerning the ex- 
ponential function. Let us think of all points with an algebraic abscissa 

as marked off on the % axis ff. >JC . We 

know that even the rational numbers, and hence, with greater reason, 
the algebraic numbers lie everywhere dense on the x axis. One might 
think at first that the algebraic numbers would exhaust the real numbers. 
But our theorem declares that this is not the case; that between the 
algebraic numbers there are infinitely many other numbers, viz. the 
'transcendental numbers; and that we have examples of them in unlimited 
quantity in algbr - no -, in log (, and in every algebraic function 
of these transcendental numbers. It will be more obvious, perhaps, if 
we write the equation in the form y = e x and draw the curve in an 
x y plane (see Fig. 118). If we now mark all the algebraic numbers on 
the x axis and on the y axis and consider all the points in the plane 
that have both an algebraic x and an algebraic y , the x y plane will be 
"densely" covered with these algebraic points. In spite of this dense 
distribution, the exponential curve y = f does not contain a single 

250 Supplement: The Theory of Assemblages. 

algebraic point except the one x = 0, y = 1 . Of all the other number 
pairs x, y which satisfy y = e? t one, at least, is transcendental. This 
course of the exponential curve is certainly a most remarkable fact. 

The full significance of these theorems which 
reveal the existence in great quantity of numbers 
which are not only not rational but which cannot 
be represented by algebraic operations upon whole 
numbers their significance for our concept of 
the number continuum is tremendous. What 
would Pythagorus have sacrificed after such a 
discovery if the irrational seemed to him to merit 
a hecatomb ! 

It is remarkable how little in general these 
questions of transcendence are grasped and assim- 
lg< " ilated, although they are so simple when one 

has thought them through. I continually have the experience, in an 
examination, that the candidate cannot even explain the notion "trans- 
cendence". I often get the answer that a transcendental number satis- 
fies no algebraic equation, which, of course, is entirely false, as the 
example x e = shows. The essential thing, that the coefficients in 
the equation must be rational, is overlooked. 

If you will think our transcendence proofs through again you will 
be able to grasp these simple elementary steps as a whole, and to make 
them permanently your own. You need to impress upon your memory 
only the Hermite integral; then everything develops itself naturally. 
I should like to emphasize the fact that in these proofs we have used 
the integral concept (or, speaking geometrically, the idea of area) as 
something in its essence thoroughly elementary, and I believe that this 
has contributed materially to the clearness of the proofs. Compare in 
this respect, the presentation in Volume I of Weber- Wellstein, or in 
my own little book, Vortrdge uber augewdhlte Fragen der Elementar- 
geometrie 1 , where, as in the older school books, the integral sign is 
avoided and its use replaced by approximate calculation of series 
developments. I think that you will admit that the proofs there are 
far less clear and easy to grasp. 

These discussions concerning the distribution of the algebraic num-' 
bers within the realm of real numbers lead us naturally to that second 
modern field to which I have often referred during these lectures, and 
which I shall now consider in some detail. 

II. The Theory of Assemblages 

The investigations of George Cantor, the founder of this theory, had 
their beginning precisely in considerations concerning the existence of 

1 Referred to p. 55. . ' 

The Power of an Assemblage. 251 

transcendental numbers 1 . They permit one to view this matter in an 
entirely new light. f 

If the -brief survey of the theory of assemblages which I shall give 
you has any special character, it is this, that I shall bring the treatment 
of concrete examples more into the foreground than is usually done 
in those very general abstract presentations which too often give this 
subject a form that is hard to grasp and even discouraging. 

1. The Power of an Assemblage 

With this end in view, let me remind you that in our earlier dis- 
cussions we have often had to do with different characteristic totalities 
of numbers which we can now call assemblages of numbers. If I confine 
myself to real numbers, these assemblages are 

1. The positive integers. 

2. The rational numbers. 

3. The algebraic numbers. 

4. All real numbers. 

Each of these assemblages contains infinitely many numbers. Our 
first question is whether, in spite of this, we cannot compare the magni- 
tude or the range of these assemblages in a definite sense, i.e., whether 
we cannot call the "infinity" of one greater than, equal to, or less than 
that of another. It is the great achievement of Cantor to have cleared 
up and answered this really quite indefinite question, by setting up 
precise concepts. Above all we have to consider his concept of power 
or cardinal number: Two assemblages have equal power (are equivalent) 
when their elements can be put into one-to-one correspondence, i.e., when 
the two assemblages can be so related to each other that to each element of 
the one there correponds one element of the other, and conversely. If such 
a mutual correspondence is not possible the two assemblages are of 
different power \ if it turns out that, no matter how one tries to set up 
a correspondence, there are always elements of one of the assemblages 
left over, this one has the greater power. 

Let us now apply this principle to the four examples given above. 
It might* seem, at first, that the power of the positive integers would 
"be smaller than that of the rational numbers, the power of these smaller 
than that of the algebraic numbers, and this finally smaller than that 
of all real numbers; for each of these assemblages arises from the pre- 
ceding by the addition of new elements. But such a conclusion would 
be too hasty. For although the power of a finite assemblage is always 
greater than the power of a part of it, this theorem is by no means valid 
for infinite assemblages. This discrepancy, after all, need not cause 

1 See Journal fur Mathematik, vol. 77 (1873), p. 258. 


Supplement: The Theory of Assemblages. 

surprise, since we are concerned in the two cases with entirely different 
fields. Let us examine a simple example which will show clearly that 
an infinite assemblage and a part of it can actually have the same 
power, the aggregate, namely, of all positive integers and that of all 
positive even integers 

1, 2, 3, 4, 5, 6, . . ., 

2, 4, 6, 8, 10, 12, 

The correspondence indicated by the double arrows is obviously of the 
sort prescribed above, in that each element of one assemblage corresponds, 
to one and only one of the other. Therefore, by Cantor's definition, the 
assemblage of the positive integers and the partial assemblage of the 
even integers have the same power. 

You see that the question as to the powers of our four assemblages 
is not so easily disposed of. The simple answer, which then appears 
the more remarkable, consists in Cantor's great discovery of 1873'. The 
three assemblages, the positive integers, the rational, and the algebraic 
numbers, have the same power', but the assemblage of all real numbers has 
another, namely, a larger power. An assemblage whose elements can be 
put into one-to-one correspondence with the series of positive integers 
(which has therefore the same power) is called denumerable. The above 

theorem can therefore be stated as 

Jl!!!!!!! follows : The assemblage of the rational 

* i wi I M I * ! I ! I ! I . 

as well as of the algebraic numbers is 
denumerable', that of all real numbers 
is not denumerable. 

Let us first give the proof for ra- 
tional numbers, which is no doubt 
familiar to some of you. Every ra- 
tional number (we shall include the ne- 
gative ones) can be expressed unique- 
ly in the form pjq, where p and q 
are integers without a common divi- 
sor, where, say, q is positive, while p' 
may also be zero or negative. In 
order to bring all these fractions p/q 

into a single series, let us mark in a p q plane all points with integral 
coordinates (p,q), so that they appear as points on a spiral path as 
shown in Fig. 119. Then we can number all these pairs (p, q) so that 
only one number will be assigned to each and all integers will be used 
(see Fig. 119). Now delete from this succession all the pairs (p, q) which 
do not satisfy the above prescription (p prime to q and q > 0) and number 



. ^ 


I * 






i , 



9 1 

? * 


C $ ; 


<r mZ 

^ H 





















ig. 1?9. 


The Power of an Assemblage. 253 

anew only those which remain (indicated in the figure by heavy points). 
We get thus a series which begins as follows: 

1 0-12 i -i -2 3 | f i . . . , 

one in which a positive integer is assigned to each rational number and 
a rational number to each positive integer. This shows that the rational 
numbers are denumerable. This arrangement of the rational numbers 

Rational number: -2 -4 -f -f ^ 7 J 2. 3 

^ \ T I I I I i i i i i i i L 

Positive integer: 7 7V J 73 6 72 Z 11 5 10 7 3 6 

Fig. 120. 

into a denumerable series requires, of course, a complete dislocation of 
their rank as to size, as is indicated in Fig. 120, where the rational 
points, laid off on the axis of abscissas, are marked with the order of 
their appearance in the artificial series. 

We come, secondly, to the algebraic numbers. I shall confine myself 
here to. real numbers, although the inclusion of complex numbers would 
not make the discussion essentially more difficult. Every real algebraic 
number satisfies a real integral equation 

which we shall assume to be irreducible, i.e., we shall omit any rational 
factors of the left-hand member, and also any common divisors of 
a l9 a lt . . ., a n . We assume also that a is always positive. Then, as 
is well known, every algebraic co satisfies but one irreducible equation 
with integral coefficients, in this normal form; and conversely, every 
such equation has as roots at most n real algebraic numbers, but perhaps 
fewer, or none at all. If, now, we could bring all these algebraic equations 
into a denumerable series we could obviously infer that their roots and 
hence all real algebraic numbers are denumerable. 

Cantor succeeded in doint this by assigning to each equation a 
definite number, its index, 

and by separating all such equations into a denumerable succession 
'of classes, according as the index N = \ , 2, 3 , . . . In no one of these 
equations can either the degree n or the absolute value of any coefficient 
exceed the finite limit N, so that, in every class, there can be only a 
finite number of equations, and hence, in particular, only a finite number 
of irreducible equations. One can easily determine the coefficients by 
trying out all possible solutions for a given N and can, in fact, write down 
at once the beginning of the series of equations for small values of N . 
Now let us consider that, for each value of the index N , the real 
roots of the finite number of corresponding irreducible equations have 

254 Supplement: The Theory of Assemblages. 

been determined, and arranged according to size. Take first the roots, 
thus ordered, belonging to index one, then those belonging to index 
two, and so on, and number them in that order. In this way we shall 
have shown, in fact, that the assemblage of real algebraic numbers is de- 
numerable, for we come in this way to every real algebraic number 
and, on the other hand, we use all the positive integers. In fact one 
could, with sufficient patience, determine say the 75 63-rd algebraic 
number of the scheme, or the position of a given algebraic number, 
however complicated. 

Here, again, our "denumeration" disturbs completely the natural 
order of the algebraic numbers, although that order is preserved among 
the numbers of like index. For example, two algebraic numbers so 
nearly equal as 2/5 and 2001/5000 have the widely separated indices 7 
and 7001 respectively; whereas ]/ 5, as root of x 2 - 5 = 0, has the 
same index, 7, as 2/5. 

Before we go over to the last example, I should like to give you 
an auxiliary theorem which will supply us with another denumerable 
assemblage, as well as with a method of proof that will be useful to us 
later on. If we have two denumerable assemblages 

a l9 a 2 , <z 3 , . . . and 6 lf ft 2 , 6 3 , . . . , 

then the assemblage of all a and all b which arises by combining these 
two is also denumerable. For one can write this assemblage as follows: 

1, &1 1 #2> &2> #3 6 3> * 

and we can at once set this into a one-to-one relation with the series of 
positive integers. Similarly, if we combine 3 , 4, . . . , or any finite number 
of denumerable assemblages, we obtain likewise a denumerable assemblage . 
But it does not seem quite so obvious, and this is to be our auxiliary 
theorem, that the combination of a denumerable infinity of denumerable 
assemblages yields also a denumerable assemblage. To prove this, let us 
denote the elements of the first assemblage by a lf a 2 , a 3 , . . ., those of 
the second by b l , 6 2 , 6 3 , . . . , those of the third by c l , c 2 , c 3 , . . . , and 
so on, and let us imagine these assemblages written under one another. 
Then we need only choose the elements of this totality according to 
successive diagonals, as indicated in the following scheme: 

The Power of an Assemblage. 255 

The resulting arrangement 

1 2 3 4 5 6 7 8 9 10 11 ... 
a^ a 2 6j 3 6 2 C] a A 6 3 c 2 d x 5 . . . 

reaches ultimately every one of the numbers a , 6 , c , . . . and brings it 
into correspondence with a definite positive integer, which proves the 
theorem. In view of this scheme one could call the process a "counting 
by diagonals'*. 

The large variety of denumerable assemblages which has thus been 
brought to our knowledge might incline us to the belief that all infinite 
assemblages are denumerable. To show that this is not true we shall 
prove the second part of Cantor's theorem, namely, that the continuum of 
all real numbers is certainly not denumerable. We shall denote it by (5^ be- 
cause we shall have occasion later to speak of multi-dimensional continua. 

(&! is defined as the totality of all finite real values x, where we 
may think of x as an abscissa on a horizontal axis. We shall first show 
that the assemblage of all inner points on the unit segment < x < 1 
has the same power as (^ . If we represent the first assemblage on the 
x axis and the second on the y axis, at right angles to it, then a one-to-one 
correspondence between them will be established by a rising monotone 
curve of the sort sketched in Fig. 121 (e.g., a branch of the curve 
y = (\lri) tan" 1 x} . It is permissible, therefore, to think of the 
assemblage of all real numbers between and 1 as standing for g t and 
we shall do this from now on. 

The proof by which I shall show you that x is not denumerable is 
the one which Cantor gave in 1891 at the meeting of the natural scientists 
in Halle. It is clearer and more susceptible of generalization than the 
one which he published in 1873. The essential thing in it is the so-called 
"diagonal process*', by which a real number is disclosed that cannot 
possibly be contained in any assumed denumerable arrangement of all 
real numbers. This is a contradiction, and (5^ cannot, therefore, be 

We write all our numbers < x < 1 as decimals and think of them 
as forming a denumerable sequence 

= 0, a a a 

= 0, b l "" 

= 0, 


where a, b, c are the digits 0, 1 , . . ., 9 in every possible choice and 
arrangement. Now we must not forget that our decimal notation is 

256 Supplement: The Theory of Assemblages. 

not uniquely definite. In fact according to our definition of equality 

we have 0.999 . . . = 1 .000 . . . , and we could write every terminating 

decimal as a non-terminating one in which all the digits, after a certain 

O 7 one, would be nines. This is one of the 

first assumptions in calculating with 
decimals (see p. 34). In order, then, 
to have a unique notation, let us 
assume that we are employing only in- 
finite, non-terminating decimals; that 
all terminating ones have been con- 
^ x verted into such as have an endless 
succession of nines; and that only in- 
finite decimals appear in our scheme 

Fig. 121. rr 


In order now to write down a decimal x which shall be different 
from every real number in the table, we fix our attention on the digits 
i, & >f c s , . . . of the diagonal of the table (hence the name of the pro- 
cess). For the first decimal place of x' we select a digit a\ different 
from a x ; for the second place a digit b' 2 different from 6 2 ; for the third 
place a digit c' 3 different from c 3 ; and so on: 

These conditions for a{ , b' 2 , c' 3 , . . . allow sufficient freedom to insure 
that x 1 is actually a proper decimal fraction, not, e.g., 0.999 . . . = 1 , 
and that it shall not terminate after a finite number of digits; in fact, 
we can select a\, b' 2 , c'%, . . . always different from 9 and 0. The x' is 
certainly different from x since they are unlike in the first decimal 
place, and two infinite decimals can be equal only when they coincide 
in every decimal place. Similarly x' ^ x 2 , on account of the second 
place; x' =j= #s> because of the third place; etc. That is, x', a proper 
decimal fraction, is different from all the numbers x lf x 2 , # 3 , . . . of the 
denumerable tabulation. Thus the promised contradiction has appeared 
and we have proved that the continuum (^ is not denumerable. 

This theorem assures us, a priori, the existence of transcendental 
numbers; for the totality of algebraic numbers was denumetable and 
could not therefore exhaust the non-denumerable continuum of all real 
numbers. But, whereas all the earlier discussions exhibited only a 
denumerable infinity of transcendental numbers, it follows here that 
the power of this assemblage is actually greater, so that it is only now 
that we get the correct general view. To be sure, those special examples 
were of service in giving life to an otherwise somewhat abstract picture 1 . 

[ l The existence of transcendental numbers was first proved by Liouville; in 
an article which appeared in 1851 in vol. 16, series 1, of the Journal des math^mati- 
ques, he gave an elementary method for constructing such numbers.] 

The Power of an Assemblage. 257 

Now that we have disposed of the one dimensional continuum it is 
very natural to inquire about the two-dimensional continuum. Every- 
body had supposed that there were more points in the plane than in the 
straight line, and it attracted much attention when Cantor showed 1 that 
the power of the two dimensional continuum ( 2 was exactly the same as 
that of the one dimensional g^. Let us take for @ 2 the square with side 
of unit length, and for (^ the unit segment (see Fig. 122). We shall 
show that the points of these two aggregates Q ^ i 

can be put into a one-to-one relation. The fact i 1 >* 

that this statement seems so paradoxical de- y 
pends probably on our difficulty in freeing our 
mental picture of a certain continuity in the 
correspondence. But the relation which we shall 
establish will be as discontinuous or, if you 
please, as inorganic as it is possible to be. It 

will disturb everything which one thinks of as F f g 122 

characteristic for the plane and the linear mani- 
fold as such, with the exception of the "power 1 '. It will be as though 
one put all the points of the square into a sack and shook them up 

The assemblage of the points of the square coincides with that of 
all pairs of decimal fractions 

x = o. a^a^a^ . . . , y = 0. b^b^ > 

all of which we shall suppose to be non-terminating. We exclude points 
on the boundary for which one of the coordinates (x, y) vanishes, i.e., 
we exclude the two sides which meet at the origin, but we include the 
other two sides. It is easy to show that this has no effect on the power. 
The fundamental idea of the Cantor proof is to combine these two 
decimal fractions into a new decimal fraction z from which one can 
obtain (x, y) again uniquely and which will take just once all the values 
< z ^ \ when the point (x, y) traverses the square once. If we then 
think of z as an abscissa, we have the desired one-to-one correpondence 
between the square ( 2 and the segment (5^, whereby the agreement 
concerning the square carries with it the inclusion of the end z = 1 
of the segment. 

One might try to effect this combination by setting 

from which one could in fact determine (x,y) uniquely by selecting 
the odd and even decimal places respectively. But there is an objection 
to this, due to the ambiguous notation for decimal fractions. This z, 
namely, would not traverse the whole of ^ when we chose for (x, y) 

i Journal fur Mathematik, vol. 84 (1878), p. 242 et seq. 

Klein, Elementary Mathematics. 17 

258 Supplement: The Theory of Assemblages. 

all possible pairs of non-terminating decimals, that is, when we traversed 
all the points of E 2 . For, although z is, to be sure, always non-termi- 
nating, there can be non-terminating values of z, such as 

z = 0. CiC 2 c 4 C Q c s . . /, 

which correspond only to a terminating x or y y in the present case to 

the values 

x = 0. c x OOO . . . , y = 0. C 2 c 4 c 6 c 8 . . . 

This difficulty is best overcome by means of a device suggested by 
J. Konig of Budapest. He thinks of the a,b,c not as individual digits 
but as complexes of digits one might call them "molecules" of the 
decimal fraction. A "molecule" consists of a single digit, different from 
zero, together with all the zeros which immediately precede it. Thus 
every non-terminating decimal must contain an infinity of molecules, 
since digits different from zero must always recur; and conversely. As 
an example, in 

x = 0.320 8007 000 302 405 ... 

we should take as molecules a^ = 3, a z = 2, a 3 08, 4 = 007, # 6 
= 0003 , a 6 = 02, 7 = 4, . . . 

Now let us suppose, in the above rule for the relation between x, y 
and z, that the a,b,c stand for such molecules. Then there will corres- 
pond uniquely to every pair (x, y) a non-terminating z which would, 
in its turn, determine x and y. But now every z breaks up into an x 
and a y each with an infinity of molecules, and each z appears therefore 
just once when (x, y) run through all possible pairs of non terminating 
decimal fractions. This means, however, that the unit segment and 
the square have been put into one-to-one correspondence, i.e., they 
have the same power. 

In an analogous way, of course, it can be shown that the continuum 
of 3 , 4 , ... dimensions has the same power as the one dimensional 
segment. It is more remarkable, however, that the continuum (S^, of 
iiifinitely many dimensions, or more exactly, of denumerably infinitely 
many dimensions, has also the same power. This infinite dimensional 
space is defined as the totality of the systems of values whi^h can be 
assumed by the denumerable infinity of variables 

when each, independently of the others, takes on all real values. This 
is really only a new form of expression for a concept that has long been 
in use in mathematics. When we talk of the totality of all power series 
or of all trigonometric series, we have, in the denumerable infinity of 
coefficients, really nothing but so many independent variables which, 
to be sure, are for purposes of calculation restricted by certain require- 
ments to ensure convergence. 

The Power of an Assemblage. 259 

Let us again confine ourselves to the "unit cube" of the (S^, i.e., 
to the totality of points which are subject to the condition < x n ^ 1 , 
and show that they can be put into one-to-one relation with the points 
of the unit segment < z ^ 1 of S^. For convenience, we exclude all 
boundary points for which one of the coordinates x m vanishes, as well 
as the end point z = 0, but admit the others. As before we start with 
the decimal fractional representation of the coordinates in K^: 

*, = 0, ^&i <* 2 3 

x 2 = O t b l b 2 b 3 . 


*3 = 0, Ct C 2 C 3 

where we assume that the decimal fractions are all written in non- 
terminating form, and furthermore that the a, b, c y . . . are "decimal 
fraction molecules 1 ' in the sense indicated above, i.e., digit complexes 
which end with a digit which is different from zero, but which is preceded 
exclusively by zeros. Now we must combine all these infinitely many 
decimal fractions into a new one which will permit recognition of its 
components; or, if we keep to the chemical figure, we wish to form such 
a loose alloy of these molecular aggregates that we can easily separate 
out the components. This is possible by means of the "diagonal process" 
which we applied before (p. 254). From the above table we get, according 
to that plan 

z = 0, a a 2 b a< 3 b 2 c^ a b Q c 2 d l a 5 . . . , 

which relates uniquely a point of x to each point of (00. Conversely 
we get in this way every point z of K lf for from the non terminating 
decimal fraction for a given z we can derive, according to the above 
given scheme, an infinity of non-terminating decimals x l , x 2 , x 3 , . . . , 
out of which this z would arise by the method indicated. We have 
succeeded therefore in setting up a one-to-one correspondence between 
the unit cube in (5^ and the unit segment in IB 

Our results thus far show that there are at least two different 
powers : 

1. That of the denumerable assemblages. 

2. That of all continua ( 1 ,( 2 >@'3> . . ., including (S^. 

The question naturally arises whether there are still larger powers. 
The answer is that one can exhibit an assemblage having a still higher 
power, not merely as a result of abstract reasoning, but one lying quite 
within the range of concepts which have long been used in mathematics. 
This aggregate is, namely: 

3. That of all possible real functions / (x) of a real variable x. 


260 Supplement: The Theory of Assemblages. 

It will be sufficient for our purpose to restrict the variable to the 
interval < x < \ . It is natural to think first of the aggregate of the 
continuous functions / (x) , but there is a remarkable theorem which 
states that the totality of all continuous functions has the same power 
as the continuum, and belongs therefore in group 2. We can reach a 
new, a higher power, only by admitting discontinuous functions of 
the most general kind imaginable, i.e., where the function value at any 
place x is entirely arbitrary and has no relation to neighboring values. 

I shall first prove the theorem concerning the aggregate of continuous 
functions. This will involve a repetition and a refinement of the con- 
siderations which we adduced (p. 206) in order to make plausible the 
possibility of developing "arbitrary" functions into trigonometric series. 
At that time I remarked: 

a) A continuous function / (x) is determined if one knows the values 
/ (r) at all rational values of r . 

b) We know now that all rational values r can be brought into a 
denumerable series r l9 r 2 , ? 3 , . . . 

c) Consequently f(x) is determined when 
one knows the denumerable infinity of quan- 
tities /(r x ), f(r 2 ), /(r s ), . . . Moreover, these 
values cannot, of course, be assumed arbit- 
rarily if we are to have a single-valued con- 
tinuous function. The assemblage then of all 
possible systems of values / (r^ , / (r 2 ) , . . . 
+x must contain a sub-assemblage whose power 
is the same as that of the assemblage of all 

Fig. 123 G 

continuous functions (see Fig. 123). 

d) Now the magnitudes / x = / (r^ , f 2 = / (r 2 ) , . . . can be considered 
as the coordinates of a (00, since they make up a denumerable infinity 
of continuously varying magnitudes. Hence, in view of the theorem 
already proved, the totality of all their possible systems of values has 
the power of the continuum. 

e) Since the assemblage of continuous functions is contained in an 
assemblage which is equivalent to the continuum, it must itself be 
equivalent to a sub-assemblage of the continuum. 

f) But it is not hard to see that, conversely, the entire continuum 
can be put into one-to-one correspondence with a part of the assemblage 
of all continuous functions. For this purpose, we need to consider only 
the functions defined by / (x) = k = const., where A; is a real parameter. 
If k traverses the continuum j then / (x) will describe an assemblage 
which is in one-to-one correspondence with (^ but which is only a part 
of the totality of all continuous functions. 

g) Now we must make use of an important general theorem of the 
theory of assemblages, the so-called theorem of equivalence, due to 

The Power of an Assemblage. 261 

F. Bernstein 1 : // each of two assemblages is equivalent to a part of the 
other then the two assemblages are equivalent. This theorem is very plau- 
sible. The proof of it would take us too far afield. 

h) According to e) and f) the continuum (5^ and the aggregate of 
all continuous functions satisfy the conditions of the theorem of equi- 
valence. They are therefore of like power, and our theorem is proved. 

Let us now go over to the proof of our first theorem, that the as- 
semblage of all possible functions that are really entirely arbitrary has 
a power higher than that of the continuum. The proof is an immediate 
application of Cantor's diagonal process. 

a) Assume the theorem to be false, i.e., that the assemblage of all 
functions can be put into one-to-one correspondence with the conti- 
nuum (]_. Suppose now, in this one-to-one relation, that the function 
/ (x, v) of % corresponds to the value x = v in lf so that, while v tra- 
verses the continuum (^ , / (x , v) represents all possible functions of x . 
We shall reduce this supposition to an absurdity by actually setting 
up a function F (x) which is different from all such functions / (x, v). 

b) For this purpose we construct the "diagonal function" of the 
tabulation of the f(x,v), i.e., that function which, for every value 
x = X Q , has that value which the assumed correspondence imposes upon 
/ (x, v) when the parameter v also has the value v = X Q , namely / (x , x ). 
Written as a function of x , this is simply the function / (x , x) . 

c) Now we construct a function .-F (x) which for every x is different 

from this f(x, x): 

F(x) 4= f(x, x) for every x. 

We can do this in the greatest variety of ways, since we admit the most 
completely discontinuous functions, whose value at any place can be 
arbitrarily determined. We might, for example, put 

F(x)=f(x,x) + i. 

d) This F(x) is actually different from every one of the functions 
f(x,v). For, if F(x) = f(x, v ) for some v = r Qt the equality would 
hold also for x = v ; that is, we should have F(r Q ) = /(>' >'o), which 
contradicts the assumption in c) concerning F (x) . 

The assumption a) that the functions f(x,v) could exhaust all func- 
'tions is thus overthrown, and our theorem is proved. 

It is interesting to compare this proof with the analogous one for 
the non-denumerability of the continuum. There we assumed the 
totality of decimal fractions arranged in a denumerable table; here we 
consider the function scheme f(x,v). The singling out there of the 
diagonal elements corresponds to the construction here of the diagonal 
function f(x, x) ; and in both cases the application was the same, namely 

1 First published in Borel's Lemons sur la Th&oiie des Fonctions, Paris, 1898, 
p. 103- 

262 Supplement: The Theory of Assemblages. 

the setting up of something new, i.e., not contained in the table, in 
the one case a decimal fraction, in the other a function. 

You can readily imagine that similar considerations could lead us 
to assemblages of ever increasing power beyond the three which we 
have already discussed. The most noteworthy thing in all these results 
is that there remain any abiding distinctions and gradations at all in 
the different infinite assemblages, notwithstanding our having subjected 
them to the most drastic treatment imaginable; treatment which 
disturbed special properties, such as order, and permitted only the 
ultimate elements, the atoms, to retain an independent existence as 
things which could be tossed about in the most arbitrary manner. And 
it is worth noting that the three gradations which we did establish were 
among things which have long been familiar in mathematics integers, 
continua, and functions. 

With this I shall close this first part of my discussion of the theory 
of assemblages, which has been devoted mainly to the concept of power. 
In a similar concrete manner, but with still greater brevity, I shall now 
tell you something about a farther chapter of this theory. 

2. Arrangement of the Elements of an Assemblage 
We shall now bring to the front just that thing which we have 
heretofore purposely neglected, the question, namely, how individual 
assemblages of the same power differ from one another by virtue of 
those relations as to the arrangement of the elements which are intrinsic 
in the assemblage. The most general one-to-one representations which 
we have admitted thus far disturbed all these relations, think only 
of the representation of the square upon the segment. I desire to 
emphasize, especially, the significance of precisely this chapter of the 
theory of assemblages. It cannot possibly be the purpose of the theory 
of assemblages to banish the differences which have long been so familiar 
in mathematics, by introducing new concepts of a most general kind. 
On the contrary, this theory can and should aid us to understand those 
differences in their deepest essence, by exhibiting their properties in 
new light. 

We shall try to make clear the different possible arrangements, by 
considering definite familiar examples. Beginning with denumerable 
assemblages, we note three examples of fundamentally different ar- 
rangement, so different that the equivalence of their powers was, as 
we saw, the result of a special and by no means obvious theorem. These 
examples are: 

1. The assemblage of all positive integers. 

2. The assemblage of all (negative and positive) integers. 

3. The assemblage of all rational numbers and that of all algebraic 

Arrangement of the Elements of an Assemblage. 263 

All these assemblages have a common property in the arrange- 
ment of their elements, which finds expression in the designation 
simply ordered, i. e., of two given elements, it is always known 
which precedes the other, or, put algebraically, which is the smaller 
and which the larger. Further, if three elements a, b, c are given, 
then, if a precedes b and b precedes c, a precedes c (if a < b and b < c 
then a <c). 

But now as to the characteristic differences. In (1), there is a first 
element (one) which preceded all the others, but no last which follows 
all the others; in (2), there is neither a first nor a last element. Both 
(1) and (2) have this in common, that every element is followed by 
another definite one, and also that every element [except the first in 
(1)] is preceded by another definite one. In contrast with this, we find 
in (3) (as we saw p. 31) that between any two elements there are always 
infinitely many others the elements are "everywhere dense", so that 
among the rational or the algebraic numbers lying between a and b 
there is neither a smallest nor a largest. The manner of arrangement 
in these three examples, the type of arrangement (Cantor's term type 
of order seems to me less expressive) is quite different, although the 
power is the same. One could raise the question here as to all the types 
of arrangement that are possible in denumerable assemblages, and that 
is what the students of the theory of assemblages actually do. 

Let us now consider assemblages having the power of the continuum. 
In the continuum x of all real numbers, we have a simply ordered 
assemblage; but in the multidimensional types ( 2 , 3 , . . . we have 
examples of an order no longer simple. In the case of S 2 , for instance, 
two relations are necessary, instead of one, to determine the mutual 
position of two points. 

The most important thing here is to analyze the concept of continuity 
for the one dimensional continuum. The recognition of the fact that 
continuity here depends on simple properties of the arrangement which 
is peculiar to C l , is the first great achievement of the theory of as- 
semblages toward the clarifying of traditional mathematical concepts. 
It was found, namely, that all the continuity properties of the ordinary 
continuum flow from its being a simply ordered assemblage with the 
following two properties: 

1 . If we separate the assemblage into two parts A , B such that every 
element belongs to one of the two parts and all the elements of A precede 
all those of B , then either A has a last element or B a first element. If we 
recall Dedekind's definition of irrational number (see p. 33 e * sec l-) 
we can express this by saying that every "cut" in our assemblage is 
produced by an actual element of the assemblage. 

2. Between any two elements of the assemblage there are always in- 
finitely many others. 

264 Supplement: The Theory of Assemblages. 

Thi.s second property is common to the continuum and the de- 
numerable assemblage of all rational numbers. It is the first property 
however that marks the distinction between the two. In the theory 
of assemblages it is customary to call all simply-ordered assemblages 
continuous if they possess the two preceding properties, for it is actually 
possible to prove for them all the thorems which hold for the continuum 
by virtue of its continuity. 

Let me remind you that these properties of continuity can be 
formulated somewhat 'differently in terms of Cantor's fundamental 
series. A fundamental series is a simply-ordered denumerable series of 
elements a lt a 2 , a 3 , . . . of an aggregate such that each element of the 
series precedes the following or each succeeds it: 

a l < a 2 < a 3 < . . . or a l > a 2 > a 3 > . . . 

An element a of the aggregate is called a limit element of the fundamental 
series if (in the first sort) every element which precedes a but no element 
which follows a is ultimately passed by elements of the fundamental 
series; and similarly for the second sort. Now if every fundamental 
series in an aggregate has a limit element, the aggregate is called closed ; 
if, conversely, every element of the aggregate is a limit element of a 
fundamental series, the aggregate is said to be dense. Now continuity, 
in the case of aggregates having the power of the continuum, consists 
essentially in the union of these two properties. 

Let me remind you incidentally that when we were discussing the 
foundations of the calculus we spoke also of another continuum, the 
continuum of Veronese, which arose from the usual one by the addition 
of actually infinitely small quantities. This continuum constitutes a 
simply-ordered assemblage in as much as the succession of any two 
elements is determinate, but it has a type of arrangement entirely 
different from that of the customary S^; even the theorem that every 
fundamental series has a limit element no longer holds in it. 

We come now to the important question as to what representations 
preserve the distinctions among the continua Si,S 2 , of ^different 
dimensions. We know, indeed, that the most general one-to-one re- 
presentation obliterates every distinction. We have here the important * 
theorem that the dimension of the continuum is invariant with respect 
to every continuous one-to-one representation, i.e., that it is impossible 
to effect a reversibly unique and continuous mapping of a ( m upon 
a ( n where m =j= n. One might be inclined to accept this theorem, 
without further ado, as self evident ; but we must recall that our naive 
intuition seemed to exclude the possibility of a reversibly unique 
mapping of ( 2 upon (5^ , and this should dispose us to caution in accepting 
its pronouncements. 

Arrangement of the Elements of an Assemblage. 265 

I shall discuss in detail only the simplest case 1 , which concerns the 
relation between the one-dimensional and the two-dimensional continua, 
and I shall then indicate the difficulties in the way of an extension to 
the most general case. We shall prove, then, that a reversibly unique, 
continuous relation between ^ and ( 2 is not possible. Every word here 
is essential. We have seen, indeed, that we may not omit continuity; 
and that reversible uniqueness may not be omitted is shown by the 
example of the "Peano curve" which is doubtless familiar to some of you. 

We shall need the following auxiliary theorem: Given two one- 
dimensional continua (^ , (&i which are mapped continuously upon each 
other so that to every element of &{ there corresponds one and but one element 
of C lf and to every element of C x there corresponds at most one element of 
(/; if, then, a, b are two elements of x to which two elements a' t b' 
in (&! actually correspond, respectively, it follows that to every element c 
of &! lying between a and b there 

will correspond an element c 1 of _i _ , - 1 -- , - 1 . - jc t 
i which lies between a 1 and V *? _ f ? _ 

(see Fig. 124). This is analogous Fig. 124. 

to the familiar theorem that a 

continuous function f(x) which takes two values a, b at the values 

% = a 7 , V must take a value c , chosen arbitrarily between a and 6, at 

some value c' between a' and V \ and it could be proved as an exact 

generalization of this theorem, by using the 

above definition of continuity. This would J - ' - ' -^ 

require one also to explain continuous map- 

ping of a continuous assemblage in a manner 

analogous to the usual definition of continu- 

ous functions, and it can be done with the 

aid of the concept of arrangement. But this 

is not the place to amplify these ideas. Fig. 125. 

We shall give our proof as follows. We 

assume that a continuous reversibly unique mapping of the one di- 
mensional segment Kj upon the square ( 2 has been effected (see 
Fig. 125). Let two elements a, b on (5^ correspond to the elements 
A, B, respectively, of 2 . Now we can join these elements A,B by 
two different paths within ( 2 , e.g., by the broken lines i,(i drawn 
in the figure. To do this, it is not necessary to presuppose any 
special properties of 2 , such as the setting up of a coordinate system; 
we need merely use the concept of double order. Each of the paths 
i and Si will be a simply-ordered one-dimensional continuum like (1, 
and because of the continuous reversibly unique relation between ( r 
and ( 2 there must correspond just one point on E t to each element of 

1 Brouwer, L. E. J. gave a proof for the general case in 1911, in volume 70, 
p. 161, of the Mathematische Annalen. 

266 Supplement: The Theory of Assemblages. 

(5 and &{ ; but to each element of S x there must correspond at most one 
on i or (i . In other words, we have precisely the conditions of the 
above lemma, and it follows that to every point c in 6^ between a and 
b there corresponds not only a point c' of 6^ but also a point ~' of &i . 
But this contradicts the assumed reversible uniqueness of the relation 
between (^ and 2 . Consequently this mapping is not possible and the 
theorem is proved. 

If one wished to extend these considerations to two arbitrary 
continua ( w , n> it would be necessary to know in advance something 
about the constitution of continua of general nature and of dimension 
1 , 2, 3 > - , w 1 , which can be embedded in & m . As soon as m y 
n^2 one can not get along merely with the concept "between' as 
we could in the simplest case above. On the contrary, one is led to very 
difficult investigations which include, among the earliest cases, the 
abstruse fundamental geometric questions concerning the most general 
continuous one-dimensional assemblage of points in the plane, questions 
which only recently have been somewhat cleared up. One of these 
interesting questions is as to when such an assemblage of points should 
be called a curve. 

I shall close with this my very special discussion of the theory of 
assemblages, in order to add a few remarks of a general nature. First, 
a word as to the general notions which Cantor had entertained concerning 
the position of the theory of assemblages with reference to geometry 
and analysis. These notions exhibit the theory of assemblages in a 
special light. The difference between the discrete magnitudes of arith- 
metic and the continuous magnitudes of geometry has always had a 
prominent place in history and in philosophical speculations. In recent 
times the discrete magnitude, as conceptually the simplest, has come 
into the foreground. According to this tendency we look upon natural 
numbers, integers, as the simplest given concepts ; we derive from them 
in the familar way, rational and irrational numbers, and we construct 
the complete apparatus for the control of geometry by means of analysis, 
namely, analytic geometry. This tendency of modern development 
can be called that of arithrrietizing geometry. The geometric 'idea of 
continuity is carried back to the idea of whole numbers. These lectures 
have, in the main, held to this direction. 

Now, as opposed to this one-sided preference for integers, Cantor 
would (as he himself told me in 1903 at the meeting of the natural 
scientists in Cassel) achieve, in the theory of assemblages, "the genuine 
fusion of arithmetic and geometry". Thus the theory of integers, on 
one hand, as well as the theory of different point continua, on the other, 
and much more, would form a homogeneous group of equally important 
chapters in a general theory of assemblages. 

Arrangement of the Elements of an Assemblage. 267 

I shall add a few general remarks concerning the relation of the theory 
of assemblages to geometry. In our discussion of assemblages we have 

1 . The power of an assemblage as something that is unchanged by 
any reversibly unique mapping. 

2. Types of order of assemblages which take account of 'the relations 
among the elements as to order. We were able here to characterize the 
notion of continuity, the different multiple arrangements or multi- 
dimensional continua, etc., so that the invariants of continuous map- 
pings found their place here. When carried over to geometry, this gives 
the branch which, since Riemann, has been called analysis situs, that most 
abstract chapter of geometry, which treats those properties of geometric 
configurations which are invariant under the most general reversibly 
unique continuous mappings. Riemann had used the word manifold 
(Mannigfaltigkeit) in a very general sense. Cantor used it also, at first, 
but replaced it later by the more convenient word assemblage (Menge). 

3. If we go over to concrete geometry we come to such differences 
as that between metric and projective geometry. It is not enough here 
to know, say, that the straight line is one-dimensional and the plane 
two-dimensional. We desire rather to construct or to compare figures, 
for which we need to use a fixed unit of measure or at least to choose 
a line in the plane, or a plane in space. In each of these concrete domains 
it is necessary, of course, to add a special set of axioms to the general 
properties of arrangement. This implies, of course, a further develop- 
ment of the theory of simply-ordered, doubly-ordered, . . ., n-tuply- 
ordered, continuous assemblages. 

This is not the place for me to go into these things in detail, 
especially since they must be taken up anyway in the succeeding vo- 
lumes of the present work. I shall merely mention literature in which 
you can inform yourselves farther. Here, above all, I should speak of 
the reports in the Mathematische Enzyklopadic : Enriques, Prinzipien 
der Geometrie (III. A. B. 1) and v. Mangoldt, Die Begriffe ,,Linie tf und 
,,Flache" (III. A. B. 2), which treat mainly the subject of axioms; also 
Dehn-Heegaard, Analysis situs (III. A. B. 3). The last article is written 
in rather abstract form. It begins with the most general formulation 
of the concepts and fundamental facts of analysis situs, as these were 
set up by Dehn himself, from which everything else is deduced then 
by pure logic. This is in direct opposition to the inductive method of 
presentation, which I always recommend. The article can be fully 
understood only by an advanced reader who has already thoroughly 
worked the subject through inductively. 

As to literature concerning the theory of aggregates, I should men- 
tion, first of all, the report made by A. Schoenflies to the Deutsche 
Mathematikervereinigung, entitled: Die Entwickelung der Lehre von 

268 Supplement: The Theory of Assemblages. 

den Punktmannigfaltigkeiten 1 . The first part appeared in volume 8 'of 
the Jahresbericht der deutschen Mathematikervereinigung; the second 
appeared recently as a second supplementary volume to the Jahres- 
bericht. This work is really a report on the entire theory of aggregates, 
in which you will find information concerning numerous details. Along- 
side of this, I would mention the first systematic textbook on the 
theory of aggregates: The Theory of Sets of Points, by W. H. Young 
and his wife, Grace Chisholm Young (whom we mentioned p. 179)- 

In concluding this discussion of the theory of assemblages we must 
again put the question which accompanies all of our lectures: How 
much of this can one use in the schools? From the standpoint of mathe- 
matical pedagogy, we must of course protest against putting such 
abstract and difficult things before the pupils too early. In order to 
give precise expression to my own view on this point, I should like to 
bring forward the biogenetic fundamental law, according to which the 
individual in his development goes through, in an abridged series, all 
the stages in the development of the species. Such thoughts have become 
today part and parcel of the general culture of everybody. Now, I think 
that instruction in mathematics, as well as in everything else, should 
follow this law, at least in general. Taking into account the native 
ability of youth, instruction should guide it slowly to higher things, 
and finally to abstract formulations; and in doing this it should follow 
the same road along which the human race has striven from its naive 
original state to higher forms of knowledge. It is necessary to formulate 
this principle frequently, for there are always people who, after the 
fashion of the mediaeval scholastics, begin their instruction with the 
most general ideas, defending this method as the "only scientific one". 
And yet this justification is based on anything but truth. To instruct 
scientifically can only mean to induce the person to think scientifically, 
but by no means to confront him, from the beginning, with cold, sci- 
entifically polished systematics. 

An essential obstacle to the spreading of such a natural and truly 
scientific method of instruction is the lack of historical knowledge which 
so often makes itself felt. In order to combat this, I have made a point 
of introducing historical remarks into my presentation. By doing this 
I trust I have made it clear to you how slowly all mathematical ideas 
have come into being; how they have nearly always appeared first in 
rather prophetic form, and only after long development have crystallized 
into the rigid form so familiar in systematic presentation. It is my 
earnest hope that this knowledge may exert a lasting influence upon 
the character of your own teaching. 

1 2 parts, Leipzig 1900 and 1908, A revision of the first half appeared in 1913 
under the title: Entwickelung dev Mengenlehre und ihrer Anwendungen; as a continu- 
ation of this, see H. Hahn: Theorie der reellen Funktionen, vol. I, Berlin, 1921. 

Index of Names. 

Abel 84, 138, 154. 
d'Alembert 103, 212. 
Archimedes 80, 209, 219, 
222, 237- 

Bachmann 39, 48. 
Ball 74. 
Baltzer 72. 
Bauer 86. 
Baumann 220. 
Berkeley 219- 
Bernoulli, Daniel 205- 
, Jacob 200. 
, Johann 200, 205, 216. 
Bernstein 261. 
Bessel 191. 
Braunmuhl 175- 
Briggs 172, 173. 
Brouwer 265- 
Budan 94. 
Biirgi 147- 
Burkhardt23,29, 191,205- 

Cantor, Georg 12, 32, 35, 
204, 221, 250, 266,267- 

Cardanus 55, 80, 134. 

Cartesius, see Descartes. 

Cauchy 84, 154, 202, 213, 
219, 228, 231, 235- 

Cavalieri 210, 214. 

Cayley 6j8, 73, 74. 

Chernac 40. 

Chisholm 179. 

Clebsch 84. 

Coble 143- 
. Copernicus 8 1 , 171. 

Coradi 198. 

Dedekind 13, 33- 
Dehn 267. 
Delambre 180, 181. 
De Moivre 153, 168. 
Descartes 81, 94. 

Dirichlet 42, 199, 202, 203, 

204, 206- 
Dyck 94. 

Enriques 55, 267. 
Eratosthenes 40. 
Eudoxus 219. 
Euklid 32, 80, 219- 
Euler 50, 56, 77, 82, 155, 

166, 200, 202, 212, 234, 


Fejer 200. 
Fermat 39, 48, 58. 
Fourier 91, 201, 204, 206. 
207, 222, 236. 

Galle 17. 

Gauss 39, 42, 50, 58, 76, 

102, 154, 181. 
Gibbs 199, 200. 
Gordan 143- 
Goursat 235. 
Grassmann 12, 58, 64. 
Gutzmer 2. 

Hahn 268. 

Hamilton 1 1, 58, 62, 73, 74. 

Hammer 175. 

Hankel 26, 56. 

Harnack 235- 

Hartenstein 99. 

Heegard 267- 

Hegel 217. 

Heiberg 80, 209- 

Hermite 238, 239, 245- 

Hilbert 13, 14, 48, 21 8, 

238, 243- 
1'Hospital 216. 

Jacobi 84. 

Kant 10. 

Kitstner 76, 210, 212. 
Kepler 208, 210. 
Kimura 74. 

Konig, J. 258. 
Kowalewski 215, 216. 
Kummer 48. 

Lacroix 235- 

Lagrange 66, 82, 83, 153, 

200, 220, 222, 234. 
Leibniz 13, 20, 56, 82, 200, 

211, 214, 215, 220, 222. 
Lie 84. 

Lindemann 238, 243, 249- 
Liouville 256. 
Liibsen 216. 
Liiroth 17. 

Maclaurin 210, 212, 234- 
Mannchen 49- 
Mangold 267. 
Markoff 236. 
Mehmke 95, 170. 
Mercator, N. 81, 150, 168. 
Michelson 198, 199- 
Minkowski 11, 39. 
Mobius 176, 177, 182. 
Molk, J. 8. 
Mollweide l8l. 
Monge 84. 

Napier, see Neper. 
Neper 81, 147,150, 172, 173- 
Netto 86. 
Newton 81, 82, 151, 168, 

210, 212, 222, 230, 233, 

Norlund 236. 

Odhner 17. 
Ohm 76. 
Ostrowski 103- 

Peano 12, 265- 
Peurbach 171. 
Picard 84, 160. 
Pitiscus 172, 174. 


Index of Names. 

Plato 80*120. 

Poisson 216. 
Pringsheim, A. 233- 
Ptolemy 170. 
Pythagorus 31, 250. 

Regiomontanus 171. 

Rhaticus 171. 

Riemann 84, 159, 202, 

Runge 86, 92, 191. 198. 

Schafheitlein 216. 
Scheffers 155, 235- 
Schellbach 222. 

| bchimmack 3, 194,223, 224. 
Schlomilch 235- 
Schoenfliess 267. 
Schubert 8. 
Seeger 189. 
Serret 86, 235- 
Shanks 237- 

Simon 5, 24, 85, 162, 221. 
Stifel 146. 
Stratton 198. 
Study 175, 181. 
Sturm, J. 94. 

Tannery, J. 8. 

Taylor 82, 153, 227, 232, 

233, 234, 236. 
Timerding 189. 

Tropjke 28,* 85, 170. 

Vega 173. 

Veronese 218, 264.' 
Vieta 25. 
Vlacq 173- - 

Weber 4, 13, 23, 29, 86, 

175, 182, 250. 
Weierstrass 33, 84, 202, 

203, 213. 
Wilbraham 198. 
Wolff, Chr. 216. 
Wolfskehl 48. 
Wiillner 217. 

Young, G. Chisholm 1 80. 
-,W. H. 268. 

Index of Contents. 

Abridged reckoning 10 et seq. 
Actually infinitely small quantities 214, 

218, 219- 
Algorithmic method, see Processes of 

growth, plan C. 
Analysis situs 267- 

Applicability and logical consistency in 
infinitesimal calculus 221. 
in the theory of fractions 29- 
complex numbers 56 58. 
irrational numbers 33. 
natural numbers 14. 
negative numbers 23 25. 
Applied mathematics 4, 15- 
Approximation, mathematics of 36. 
Archimedes, axiom of 218. 
Arithmetization 266. 
Arrangement within an assemblage 262. 
Assemblage of continuous and real func- 
tions 206, 259-261. 

of algebraic and transcendental 
numbers 250, 254 256. 

Branch points 107, 109- 

Calculating machines 17 21. 

and formal rules of operation 21, 22. 
Cardinal number 251- 

Casus irreducibilis of the cubic equation 

Circular functions: 

analogy with hyperbolic functions 

see also trigonometric functions. 
Closed fundamental series 264. 
Complex numbers, higher 5875. 
Consistency, proofs of 13, 25, 57. 

and applicability : 

of infinitesimal calculus 221. 
of the theory of fractions 30. 

complex numbers 55 58. 

irrational numbers 34. 

natural numbers 14. 

negative numbers 23. 

Constructions with ruler and compasses 


Continued fractions 4244. 
Continuity, analysis of, based on theory 

of assemblages 263 266. 
Curriculum proposals, the Meran 16. 
Cut, after Dedekind 33. 
Cyclometric functions: 

definition of, by means of quadra- 
ture of the circle 163 168. , 
Cyclotomic numbers 47- 

Decimal system 6, 9, 20. 
Dense 31, 249, 263, 264. 
Dcnumerability of algebraic numbers 
i 253 et seq. 

rational numbers 252 et seq. 

a denumerable infinity of de- 
numerable assemblages 254. 

Derivative calculus 220, 234. 
Development of infinitesimal calculus 


Diagonal process 254, 259, 261. 
Differences, calculus of 228, 230232. 
Differentials, calculation with: 

naive intuitional direction 208 210. 

direction of mathematics of ap- 
proximation 215, 216. 

formal direction 215. 

speculative direction 214, 216, 217. 
Dimension, in variance of the of a 

continuum by reversibly unique 

mapping 264, 265- 
Discriminant curve of the quadratic and 

cubic equation 92. 

surface of the biquadratic equation 

Equations : 

cyclotomic 50. 

pure 110-115, 131-134. 

reciprocal 51. 

of fifth degree 141-142. 

the dihedral 115 120, 126. 

the tetrahedral 120130. 


Index of Contents. 

the octahedral 120130. 
the icosahedral 120130. 
Equivalence, of assemblages 251262. 
, theorem of 260. 
Exhaustion, method of 209. 
Exponential function: 

definition by quadrature of hyper- 
bola 149 et seq., 156-157- 
general , and e w 158 159, 

series for e x 152. 

function -theoretic discussion of 156 
et seq. 

Fermat, great theorem of 4649- 
Formal mathematics 24, 26, 29, 56. 
Foundations of arithmetic: 
by means of intuition 11. 

formalism 13. 

logic 11. 

theory of point sets 12. 
Fourier 's series, see trigonometric series. 

integral 207- 
Function, notion of: 

analytic function 200201. 

arbitrary function 200. 

relation of the two in complex region 

discontinuous real functions 204. 
Functions, assemblage of continuous 

and real 206, 261262. 
Fundamental laws of addition and 

multiplication 810. 

logical foundation 1016. 

consistency 13 et seq. 

regions on the sphere 111 114, 

series, Cantor's 264. 

theorem of algebra 101 104. 
Fractions, changing common into deci- 
mal 40. 

Gamma function 239. 
Graphical methods for equations in the 
complex field 102133. 

determining the real solutions 
of equations 87 101. 

Historical excursus on : 

relations between differential cal- 
culus and the calculus of finite 
differences 232235. 

exponential function and logarithm 

the notion of function 200207. 

infinitesimal calculus 207 223. 

imaginary numbers 55, 75 76. 

irrational numbers 31 34. 

negative numbers 25 27. 

Taylor's theorem 233 234- 

transcendence of e and n 237 238. 

trigonometric series 205207. 

trigonometric tables and logarithmic 
tables 170-174. 

the modern development and the 
general structure of mathematics 

Homogeneous variables in function 

theory 106-108. 
Hyperbolic functions 164166. 

analogy with circular functions 166. 

fundamental function for 166. 

Impossibility, proofs of: 

general 51- 

construction of regular heptagon 
with ruler and compasses 51 55- 

trisection of an angle 114. 
Induction, mathematical 11. 
Infinitesimal calculus, invention and 

development of 207 et seq. 
Instruction, reform in 5. 
Interpolation : 

by means of polynomials after 
Lagrange 229- 
Newton 229232. 

trigonometric 190193- 
Interpolation parabolas 229- 
Investigation, mathematical 208. 
Irreducibility : ^ 

function-theoretic 113114. 

number-theoretic 52. 

Lagrange f s interpolation formula 229. 
Limit, method of 211-214. 
Logarithm : 

base of the natural 150151- 
calculation of 148 et seq., 1 72 et seq. 
definition of the natural by means 
of quadrature of the hyperbola 
149, 156. 

difference equation for the 148. 
function -theoretic discussion of 


uniformization by means of 133, 

Mean-value theorem of differential 
calculus 213-214; extension of 
same 231 et seq. 

Index of Contents. 


Newton's interpolation formula 229 

to 232. 
Nomographic scales for: 

order curves 89, 94. 

class curves 90, 95- 
Non-denumerability of the continuum 


Non-Archimedean number system 218. 
Normal class curve of biquadratic 

equation 9698. 

curves as: 

class curves 9093, 95, 97- 
order curves 8990, 94. 

equations of the regular bodies: 
solution by separation and series 


uniformization 133 138. 

- - radicals 138141. 

reduction of general equations to 

normal equations 141 143- 
Number, assemblage of continuous and 

real numbers 250, 251253- 
, notion of 10. 

, transition from, to measure 28. 
- pair 28, 56. 

scale 23, 26, 31. 

Order, types of 263- 
Osculating parabolas 224 226. 
limiting form of 227. 

Peano curve 265- 
Perception, inner n. 

and logic 11. 
Philologists, relation to 2. 
Picard's theorem 160. 

Point, the infinitely distant of the 

complex plane 105- 
Point lattice 43- 
Power of the continuum of a de- 

numerable infinity of dimensions 258 . 

of a finite number of dimensions 

of ar? assemblage 251262. 

the assemblage of all real func- 

tions 261. 

continuous functions 260. 
Precision, mathematics of 36. 
Prime numbers, existence of infinitely 

many 40. 

factor tables 40. 
Principle of permanence 26. 
Process of growth of mathematics: 

Plan A. Separating methods and 

disciplines; logical direction 75. 
Klein, Elementary Mathematics. 

Plan B. Fusing methods and dis- 
ciplines; intuitive direction 77. 
Plan C. Algorithmic process; for- 
mal direction 79. 
Psychologic moments in teaching 4, 10, 

16, 28, 30, 34, 268. 
Pythagorean numbers 44. 

Quaternion 6075- 

scalar part of 60. 

vector part of 60. 

tensor of 63, 66, 72. 

versor of 72. 

Rational, in the sense of mathematics of 

approximation 36. 
Reform, the Basel aims toward 2. 


the beginnings of infinitesimal cal- 
culus in school instruction 223; 
see also curriculum proposals and 
reform in instruction. 


Dresden for training teachers 2. 

Regular bodies, groups of 120124. 
Rieman surfaces 105 110. 

sphere 105110. 
Rotation of space 73- 

and expansion of space 67 73. 

School instruction: 

treatment of fractions 27. 

rrational numbers 37. 

complex numbers 75. 

the pendulum 187-190. 
exposition of the formal rules of 

operation 10. 

introduction of negative numbers 22, 28. 
notion of function 205- 
infinitesimal calculus 221 et seq. 
exponent and logarithm 144146, 

operations with natural numbers 

trigonometric solution of cubic 

equation 134137- 
transition to operations with letters 8. 
uniformization of the pure equa- 
tion by means of the logarithm 

number-theoretic considerations 


mathematics, contents of 4 



Index of Contents. 

Signs, rule of 24. 
quasi proof for 26. 
Space perception 35. 
Square root expressions: 

significance of for constructions with 
ruler and compasses 50. 

classification of 53- 
Sturm's theorem, geometrical equivalent 

of 94. 
Style of mathematical presentation 84. 

Taylor's formula 223, 233- 

analogy with Newton's interpolation 
formula 232 et seq. 

remainder term 226, 231. 
Teachers, academic education of 1. 
, academic and normal school training 

of 7. 

Tensor 63, 66, 70, 72. 
Terminology, different in the schools : 

algebraic numbers 23. 

arithmetic 3. 

relative numbers 23. 
, misleading in: 

algebraically soluble 140. 

irreducible 136. 

root 140. 

Maclaurin's series 224. 
Threshold of perception 35. 
Transcendence of e 237 243. 
- of n 243-249. 

Triangle, notion of in spherical tri- 

elementary 175. 

proper and improper 181 182. 

with Mdbius 176, 177, 182-183- 

with Study l8l. 

triangular membranes 183 186. 
Trigonometry, spherical 175 186. 

its place in geometry of hyperspace 
178-182. . 

supplementary relations of 1 83 1 86. 
Trigonometric functions, see circular 

Trigonometric functions: 

calculation of 170 174 

definition by means of quadrature 
of circle 162 et seq. 

complex fundamental function for 
165 et seq. 

real fundamental function for 166 
et seq. 

function - theoretic discussion of 

application of to spherical trigono- 
metry 175 186. 

application of to oscillations of 
pendulum 186190. 

application of to representation of 
periodic functions 190200; see 
also trigonometric series. 

series 190200. 
Gibb's phenomenon 199- 
approximating curves 194196. 
convergence, proof of 196198. 
trigonometric interpolation 190-193. 
behavior at discontinuities 197 et seq. 

Uniformization 133, 138. 

by means of logarithm 134, 159- 

Vector 60, 63-65- 
Versor 72.